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CBSE Questions for Class 12 Commerce Applied Mathematics Definite Integrals Quiz 9 - MCQExams.com
CBSE
Class 12 Commerce Applied Mathematics
Definite Integrals
Quiz 9
Let
I
1
=
∫
2
1
1
√
1
+
x
2
d
x
and
I
2
=
∫
2
1
1
x
d
x
. Then
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0%
I
1
>
I
2
0%
I
2
>
I
1
0%
I
1
=
I
2
0%
I
1
>
2
I
2
Explanation
I
1
=
∫
2
1
1
√
1
+
x
2
d
x
=
(
log
(
x
+
√
1
+
x
2
)
)
2
1
=
log
(
2
+
√
5
)
−
log
(
1
+
√
2
)
=
log
(
(
2
+
√
5
)
(
1
+
√
2
)
)
I
2
=
∫
2
1
1
x
d
x
=
(
log
x
)
2
1
=
log
2
−
log
1
=
log
2
∴
The value (s) of
\displaystyle \int_{0}^{1}\frac{x^{4}\left ( 1-x \right )^{4}}{1+x^{2}} dx
is (are)
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\displaystyle \frac{22}{7}-\pi
0%
\displaystyle \frac{2}{105}
0%
0
0%
\displaystyle \frac{71}{15}-\frac{3\pi}{2}
Explanation
Let
\displaystyle I=\int _{ 0 }^{ 1 }{ \frac { { x }^{ 4 }{ \left( 1-x \right) }^{ 4 } }{ 1+{ x }^{ 2 } } } dx
\displaystyle =\int _{ 0 }^{ 1 }{ \frac { \left( { x }^{ 4 }-1 \right) { \left( 1-x \right) }^{ 4 }+{ \left( 1-x \right) }^{ 4 } }{ \left( 1+{ x }^{ 2 } \right) } dx }
\displaystyle =\int _{ 0 }^{ 1 }{ \left( { x }^{ 2 }-1 \right) } { \left( 1-x \right) }^{ 4 }dx+\int _{ 0 }^{ 1 }{ \frac { { \left( 1+{ x }^{ 2 }-2x \right) }^{ 2 } }{ \left( 1+{ x }^{ 2 } \right) } dx } \\
\displaystyle =\int _{ 0 }^{ 1 }{ \left\{ \left( { x }^{ 2 }-1 \right) { \left( 1-x \right) }^{ 4 }+\left( 1+{ x }^{ 2 } \right) -4x+\frac { { 4x }^{ 2 } }{ \left( 1+{ x }^{ 2 } \right) } \right\} dx }
\displaystyle =\int _{ 0 }^{ 1 }{ \left( \left( { x }^{ 2 }-1 \right) { \left( 1-x \right) }^{ 4 }+\left( 1+{ x }^{ 2 } \right) -4x+4\frac { 4 }{ 1-{ x }^{ 2 } } \right) dx }
\displaystyle =\int _{ 0 }^{ 1 }{ \left( { x }^{ 6 }-{ 4x }^{ 5 }+{ 5x }^{ 4 }-{ 4x }^{ 2 }+4-\frac { 4 }{ 1+{ x }^{ 2 } } \right) dx }
\displaystyle =\left[ \frac { { x }^{ 7 } }{ 7 } -\frac { { 4x }^{ 6 } }{ 6 } +\frac { { 5x }^{ 5 } }{ 5 } -\frac { { 4x }^{ 3 } }{ 3 } +4x-4\tan ^{ -1 }{ x } \right] _{ 0 }^{ 1 }
\displaystyle =\frac { 1 }{ 7 } -\frac { 4 }{ 6 } +\frac { 5 }{ 5 } -\frac { 4 }{ 3 } +4-4\left( \frac { \pi }{ 4 } -0 \right) =\frac { 22 }{ 7 } -\pi
If
\displaystyle I= \int_{0}^{1}\frac{x dx}{8+x^{3}}
then the smallest interval in which
I
lies is
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\displaystyle \left ( 0,\frac{1}{8} \right )
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\displaystyle \left ( 0,\frac{1}{9} \right )
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\displaystyle \left ( 0,\frac{1}{10} \right )
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\displaystyle \left ( 0,\frac{1}{7} \right )
Explanation
\displaystyle I=\int _{ 0 }^{ 1 }{ \frac { x }{ 8+{ x }^{ 3 } } } dx=\int _{ 0 }^{ 1 }{ \frac { x }{ \left( x+2 \right) \left( { x }^{ 2 }-2x+4 \right) } } dx
\displaystyle =\int _{ 0 }^{ 1 }{ \left( \frac { x+2 }{ 6\left( { x }^{ 2 }-2x+4 \right) } -\frac { 1 }{ 6\left( x+2 \right) } \right) } dx
\displaystyle =\frac { 1 }{ 6 } \int _{ 0 }^{ 1 }{ \left( \frac { 2x-2 }{ 2\left( { x }^{ 2 }-2x+4 \right) } +\frac { 3 }{ 6{ x }^{ 2 }-2x+4 } \right) dx } -\frac { 1 }{ 6 } \int { \frac { 1 }{ x+2 } } dx
\displaystyle =\frac { 1 }{ 2 } \int _{ 0 }^{ 1 }{ \frac { 2x-2 }{ { x }^{ 2 }-2x+4 } } dx+\frac { 1 }{ 2 } \int _{ 0 }^{ 1 }{ \frac { 1 }{ { x }^{ 2 }-2x+4 } } -\frac { 1 }{ 6 } \int _{ 0 }^{ 1 }{ \frac { 1 }{ x+2 } } dx
\displaystyle =\left[ \frac { 1 }{ 12 } \log { \left( { x }^{ 2 }-2x+4 \right) } -\frac { 1 }{ 6 } \log { \left( x+2 \right) } \right] +\frac { \tan ^{ -1 }{ \left( \frac { x-1 }{ \sqrt { 3 } } \right) } }{ 2\sqrt { 3 } }
\displaystyle =\frac { 1 }{ 12 } \log { \left( 3 \right) } -\frac { 1 }{ 6 } \log { \left( 3 \right) } +\frac { \tan ^{ -1 }{ \left( \frac { 2 }{ \sqrt { 3 } } \right) } }{ 2\sqrt { 3 } }
\displaystyle -\frac { 1 }{ 12 } \log { \left( 4 \right) -\frac { 1 }{ 6 } \log { \left( 2 \right) } } +\frac { \tan ^{ -1 }{ \left( \frac { -1 }{ \sqrt { 3 } } \right) } }{ 2\sqrt { 3 } }
And this lies between
\displaystyle \left( 0,\frac { 1 }{ 9 } \right)
If
\displaystyle I_{n} =\int_{0}^{\frac{\pi }{4}}\tan ^{n}xdx
then
\displaystyle \frac{1}{I_{2}+I_{4}},\frac{1}{I_{3}+I_{5}},\frac{1}{I_{4}+I_{6}}
are in?
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A.P
0%
H.P
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G.P
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None of these
Explanation
\displaystyle I_{n}= \int_{0}^{\pi /4}\tan ^{n-2}x\tan ^{2}xdx
\displaystyle= \int_{0}^{\pi /4}\tan ^{n-2}x\left ( \sec ^{2}x-1 \right )dx=\int_{0}^{\pi /4}\tan ^{n-2}x \sec ^{2}x dx -\int_{0}^{\pi /4}\tan ^{n-2}xdx
\displaystyle I_{n}= \left [ \frac{\tan ^{n-1}x}{n-1} \right ]_{0}^{\pi /4}-I_{n-2}
\displaystyle \therefore I_{n}+I_{n-2}= \frac{1}{n-1}
Substitute
n=4,5,6, ....
we get
\displaystyle I_{4}+I_{2}, I_{5}+I_{3}, I_{6}+I_{4},....
are respectively
\displaystyle \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\cdots
which are in H.P. and hence their reciprocals are in A.P.
If
\displaystyle I_{t}=\int_{0}^{\dfrac{\pi }{2}}\frac{\sin^{2}tx}{\sin^{2}x}dx
then ,
I_{1},I_{2},I_{3}
are in
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A.P.
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H.P.
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G.P.
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None of these
Explanation
Consider
\displaystyle I_{t+2}+I_{t}-2I_{t+1}
\displaystyle \int_{0}^{\pi /2}\frac{\left ( \sin ^{2}tx+\sin^{2}\left ( t+2 \right )x-2\sin ^{2}\left ( t+1 \right )x \right )dx}{\sin ^{2}x}
\displaystyle \int_{0}^{\pi /2}\frac{\left ( \sin ^{2}\left ( t+2 \right )x-\sin^{2}\left ( t+1 \right )x-\sin ^{2}\left ( t+1 \right )x+\sin^{2}tx \right )dx}{\sin ^{2}x}
\displaystyle \int_{0}^{\pi /2}\frac{\left [ \sin \left ( 2t+3 \right )x-\sin \left ( 2t+1 \right )x \right ]dx}{\sin x}
\displaystyle =2\int_{0}^{\pi /2}\cos \left ( 2t+2 \right )x dx=2\left ( \frac{\sin \left ( 2t+2 \right )x}{2\left ( t+1 \right )} \right )^{\pi /2}_{0}
\displaystyle = \frac{1}{\left ( t+1 \right )}\left ( 0 \right )=0
\displaystyle \therefore I_{t},I_{t+1},I_{t+2}\epsilon A.P.
Short Cut Method :
Substitute
t=1, 2, 3
in given integral we get
\displaystyle I_{1}= \frac{\pi}{2},I_{2}=\pi , I_{3}=\frac{3\pi }{2}
\displaystyle I_{1},I_{2}, I_{3}\epsilon A.P.
The value of
\displaystyle \int_{0}^{\pi /2}\sin \theta \log \left ( \sin \theta \right )\:d\theta
equals
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\displaystyle \log_{e}\left ( \frac{1}{e} \right )
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\displaystyle \log _{2}e
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\displaystyle \log_{e}{2}-1
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\displaystyle \log_{e}\left ( \frac{e}{2} \right )
Explanation
Let
I=\displaystyle\int_{0}^{\displaystyle\frac{\pi}{2}}\sin \theta \log(\sin \theta )d\theta =\displaystyle \frac{1}{2}\int_{0}^{\displaystyle \frac{\pi}{2}}\sin \theta \log(\sin^{2} \theta )d\theta \left ( \because \log_{e}a^{m}=m\log_{e}a \right )
=\displaystyle\frac{1}{2}\int_{0}^{\displaystyle \frac{\pi}{2}}\sin \theta \log(1-\cos^{2}\theta)d\theta
Take
t=\cos\theta
\Rightarrow \sin \theta d\theta=-dt
\theta=0,t=1
and
\theta =\displaystyle\frac{\pi}{2},t=0
Then,
I=\displaystyle\frac{1}{2}\int_{0}^{1}\log(1-t^{2})dt
=-\displaystyle \frac{1}{2}\left [ \int_{0}^{1}\left ( t^{2}+\frac{t^{4}}{2}+\frac{t^{6}}{3}+...\infty \right )dt \right ]
=-\displaystyle\frac{1}{2}\left [\displaystyle\frac{1}{3}+\frac{1}{2\cdot5}+\frac{1}{7\cdot3}+...\infty \right ]
=-\left [\displaystyle\frac{1}{2.3}+\frac{1}{4\cdot5}+\frac{1}{6\cdot7}+....\infty \right ]
=-\left [\left (\displaystyle\frac{1}{2}-\frac{1}{3}\right )+\left (\displaystyle\frac{1}{4}-\displaystyle\frac{1}{5} \right )+\left (\displaystyle\frac{1}{6}-\displaystyle \frac{1}{7} \right )+....\infty\right ]
=-\displaystyle\frac{1}{2}+\frac{1}{3}-\displaystyle\frac{1}{4}+\frac{1}{5}-\displaystyle \frac{1}{6}+\frac{1}{7}+....\infty
=\left(1-\displaystyle\frac{1}{2}+\frac{1}{3}-\displaystyle\frac{1}{4}+\frac{1}{5}-\displaystyle \frac{1}{6}+\frac{1}{7}+....\infty\right)-1
=\log_{e}2-1
=\log_{e}2-\log e=\log_{e}\left (\displaystyle\frac{2}{e} \right )
Ans: C
If
f(x) =\displaystyle \underset{1}{\overset{x}{\int}} \dfrac{\tan^{-1} t}{t} dt \, \, \forall \in R
, then the value of
f(e^2) - f \left (\dfrac{1}{e^2}\right)
is
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0
0%
\dfrac{\pi}{2}
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\pi
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2 \pi
The value(s) of
\int _{ 0 }^{ 1 }{ \cfrac { { x }^{ 4 }{ \left( 1-x \right) }^{ 4 } }{ 1+{ x }^{ 2 } } } dx
is (are)
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\cfrac{22}{7}-\pi
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\cfrac{2}{105}
0%
0
0%
\cfrac{71}{15}-\cfrac{3\pi}{2}
If
\displaystyle 2f(x)+f(-x)=\frac{1}{x}\sin{\left(x-\frac{1}{x}\right)}
, then the value of
\displaystyle\int_{\frac{1}{e}}^{e}{f(x)dx}
, is
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1
0%
0
0%
e
0%
-1
Explanation
Since,
\displaystyle 2f(x)+f(-x)=\frac{1}{x}\sin{\left(x-\frac{1}{x}\right)}
...(i)
\displaystyle\therefore 2f(-x)+f(x)=\frac{1}{x}\sin{\left(x-\frac{1}{x}\right)}
(replace
x
by
-x
) ...(ii)
\implies f(x)=f(-x)
[subtracting equations (i) and (ii)]
\displaystyle\therefore 3f(x)=\frac{1}{x}\sin{\left(x-\frac{1}{x}\right)}
Hence,
\displaystyle I=\int_{\frac{1}{e}}^{e}{f(x)dx}=\frac{1}{3}\int_{\frac{1}{e}}^{e}{\frac{1}{x}\sin{\left(x-\frac{1}{x}\right)}dx}
Now, put
\displaystyle x=\frac{1}{t}\Rightarrow\displaystyle dx=-\frac{1}{t^2}dt
\displaystyle\therefore I=\frac{1}{3}\int_{e}^{\frac{1}{e}}{t\sin{\left(\frac{1}{t}-t\right)}.\left(-\frac{1}{t^2}\right)dt}
\displaystyle=\frac{1}{3}\int_{e}^{\frac{1}{e}}{\frac{1}{t}\sin{\left(t-\frac{1}{t}\right)}dt}
\displaystyle=-\frac{1}{3}\int_{\frac{1}{e}}^{e}{\frac{1}{t}\sin{\left(t-\frac{1}{t}\right)}dt}
\Rightarrow I=-I\Rightarrow 2I=0\Rightarrow I=0
\displaystyle\therefore\int_{\frac{1}{e}}^{e}{f(x)dx}=0
.
If
\displaystyle I= \int_{1/\pi }^{\pi }\frac{1}{x}\cdot \sin \left ( x-\frac{1}{x} \right )dx
then I is equal to
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0
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\displaystyle \pi
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\displaystyle \pi -\frac{1}{\pi }
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\displaystyle \pi +\frac{1}{\pi }
Explanation
Let
\displaystyle I=\int _{ \dfrac { 1 }{ \pi } }^{ \pi } \frac { 1 }{ x } \cdot \sin \left( x-\dfrac { 1 }{ x } \right) dx
Substitute
\displaystyle x=\dfrac { 1 }{ t }
\displaystyle \therefore I=\int _{ \pi }^{ \dfrac { 1 }{ \pi } }{ t\sin { \left( \dfrac { 1 }{ t } -t \right) } \left( -\dfrac { 1 }{ { t }^{ 2 } } \right) dt }
\displaystyle =-\int _{ \dfrac { 1 }{ \pi } }^{ \pi }{ \dfrac { 1 }{ t } \sin { \left( t-\dfrac { 1 }{ t } \right) dt } } -I\\ \Rightarrow 2I=0\Rightarrow I=0
If
x
satisfies the equation
\displaystyle\left(\int_0^1{\frac{dt}{t^2+2t\cos{\alpha}+1}}\right)x^2-\left(\int_{-3}^3{\frac{t^2\sin{2t}}{t^2+1}dt}\right)x-2=0
for
(0<\alpha<\pi)
then the value of
x
is?
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\displaystyle\pm\sqrt{\frac{\alpha}{2\sin{\alpha}}}
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\displaystyle\pm\sqrt{\frac{2\sin{\alpha}}{\alpha}}
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\displaystyle\pm\sqrt{\frac{\alpha}{\sin{\alpha}}}
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\displaystyle\pm2\sqrt{\frac{\sin{\alpha}}{\alpha}}
Explanation
\displaystyle \left( \int _{ 0 }^{ 1 }{ \frac { dt }{ t^{ 2 }+2t\cos { \alpha } +1 } } \right) x^{ 2 }-\left( \int _{ -3 }^{ 3 }{ \frac { t^{ 2 }\sin { 2t } }{ t^{ 2 }+1 } dt } \right) x-2=0
Here,
\displaystyle \int _{ -3 }^{ 3 }{ \frac { t^{ 2 }\sin { 2t } }{ t^{ 2 }+1 } dt } =0 \left(\because f(t)=\frac { t^{ 2 }\sin { 2t } }{ t^{ 2 }+1 } \text{ is odd}\right)
So, the equation becomes
\displaystyle \left( \int _{ 0 }^{ 1 }{ \frac { dt }{ t^{ 2 }+2t\cos { \alpha } +1 } } \right) x^{ 2 }-2=0
....(1)
Now,
\displaystyle \int _{ 0 }^{ 1 }{ \frac { dt }{ t^{ 2 }+2t\cos { \alpha } +1 } } =\int _{ 0 }^{ 1 }{ \frac { dt }{ (t+\cos { \alpha } )^{ 2 }+\sin ^{ 2 }{ \alpha } } }
=\displaystyle \frac { 1 }{ \sin { \alpha } } \left[{ \tan ^{ -1 }{ \left(\frac { t+\cos { \alpha} }{ \sin { \alpha } } \right) } }\right]_{ 0 }^{ 1 }
\displaystyle =\frac { 1 }{ \sin { \alpha } } \left[\tan ^{ -1 }{\cot {\dfrac{\alpha}2 } -\tan ^{ -1 }{ \cot { \alpha } } }\right]
\displaystyle =\frac { 1 }{ \sin { \alpha } } \left[\frac { \pi }{ 2 }-\frac { \alpha }{ 2 } -\frac { \pi }{ 2 } +\alpha\right]
\displaystyle =\frac { \alpha }{ 2\sin { \alpha } }
So, equation (1) becomes
\displaystyle\frac { \alpha }{ 2\sin { \alpha } } x^{ 2 }-2=0
\Rightarrow \displaystyle x=\pm 2 \sqrt {\frac{\sin \alpha}{\alpha}}
Hence, option D.
The tangent to the graph of the function
\displaystyle y = f(x)
at the point with abscissa x = a forms with the x-axis an angle of
\displaystyle \pi/3
and at the point with abscissa x = b at an angle of
\displaystyle \pi/4
, then the value of the integral,
\displaystyle \int_{a}^{b} f'(x).f''(x)dx
is equal to
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1
0%
0
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\displaystyle -\sqrt 3
0%
-1
Explanation
Given , at
x=a , \frac{dy}{dx}=\tan {\frac{\pi}{3}}=\sqrt{3}
\Rightarrow f'(a)=\sqrt{3}
Also, at
x=b, \frac{dy}{dx}=\tan {\frac{\pi}{4}}=1
\Rightarrow f'(b)=1
Now,
\displaystyle \int_{a}^{b} f'(x).f''(x)dx
Put
f'(x)=t \Rightarrow f''(x)dx=dt
=\displaystyle \int_{\sqrt{3}}^{1} t dt
=-\displaystyle \int_{1}^{\sqrt{3}} t dt
\displaystyle =-[\frac{t^{2}}{2}]_{1}^{\sqrt{3}}
=-1
Let
\displaystyle F\left ( x \right )=f\left ( x \right )+f\left ( \frac{1}{x} \right )
where
\displaystyle f\left ( x \right )=\int_{1}^{x}\frac{\log t}{1+t}dt
Then
F(e)
is equal to?
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0%
1
0%
2
0%
1/2
0%
0
Explanation
\displaystyle F\left ( x \right )=\int_{1}^{x}\frac{\ln t}{1+t}dt+\int_{1}^{1/x}\frac{\ln t}{1+t}dt
\displaystyle F\left ( x \right )=\int_{1}^{x}\left ( \frac{\ln t}{1+t}+\frac{\ln t}{\left ( 1+t \right )t} \right )dt=\int_{1}^{t}\frac{\ln t}{t}dt=\frac{1}{2}\left ( \ln x \right )^{2}
F\left ( e \right )=1/2
The value of
\displaystyle \int _{ 0 }^{ \pi /2 }{ \frac { d\theta }{ 5+3\cos { \theta } } }
is?
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\displaystyle \tan ^{ -1 }{ \frac { 1 }{ 2 } }
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\displaystyle \tan ^{ -1 }{ \frac { 1 }{ 3 } }
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\displaystyle \frac{1}{2}\tan ^{ -1 }{ \frac { 1 }{ 2 } }
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\displaystyle \frac{1}{3}\tan ^{ -1 }{ \frac { 1 }{ 3 } }
Explanation
Let
I=\displaystyle \int _{ 0 }^{ \pi /2 }{ \frac { d\theta }{ 5+3\cos { \theta } } } =\int _{ 0 }^{ \pi /2 }{ \frac { d\theta }{ 5+3.\frac { 1-\tan ^{ 2 }{ \frac { \theta }{ 2 } } }{ 1+\tan ^{ 2 }{ \frac { \theta }{ 2 } } } } }
\displaystyle =\int _{ 0 }^{ \pi /2 }{ \frac { \sec ^{ 2 }{ \frac { \theta }{ 2d\theta } } }{ 8+2\tan ^{ 2 }{ \frac { \theta }{ 2 } } } }
\displaystyle =\int _{ 0 }^{ \pi /2 }{ \frac { \frac { 1 }{ 2 } \sec ^{ 2 }{ \left( \frac { \theta }{ 2 } \right) } d\theta }{ 4+\tan ^{ 2 }{ \left( \frac { \theta }{ 2 } \right) } } } =\int _{ 0 }^{ 1 }{ \frac { dt }{ 4+{ t }^{ 2 } } }
[
Put
\displaystyle \tan { \frac { \theta }{ 2 } } =t
So,
\dfrac { 1 }{ 2 } \sec ^{ 2 }{ \dfrac { \theta }{ 2 } } d\theta =dt]
\therefore I=\displaystyle =\frac { 1 }{ 2 } \left[ \tan ^{ -1 }{ \frac { t }{ 2 } } \right] _{ 0 }^{ 1 }
\displaystyle =\frac { 1 }{ 2 } \left( \tan ^{ -1 }{ \frac { 1 }{ 2 } } -\tan ^{ -1 }{ 0 } \right)
= \displaystyle \frac { 1 }{ 2 } \tan ^{ -1 }{ \dfrac { 1 }{ 2 } }
Evaluate :
\displaystyle \underset{0}{\overset{\infty}{\int}} \dfrac{dx}{(1 + x^2)^4}
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\dfrac{\pi}{32}
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\dfrac{3 \pi}{32}
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\dfrac{5 \pi}{32}
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\dfrac{7 \pi}{32}
Let
f:R\rightarrow R^{+}
and
I_{I}=\int^{k}_{1-k}\,xf(x(1-x))\,dx,I_2=\int^{k}_{1-k}f(x(1-x))\,dx
where
2k-1>0
. Then
\dfrac{I_I}{I_2}
is
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0%
2
0%
k
0%
1/2
0%
1
The value of
\int _{ 0 }^{ \infty }{ \cfrac { \log { x } }{ { a }^{ 2 }+{ x }^{ 2 } } }
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\cfrac{2\pi\log{a}}{a}
0%
\cfrac{\pi\log{a}}{2a}
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\pi \log{a}
0%
0
\displaystyle \int_{-1}^{1} \cot^{-1} \left(\dfrac{x+x^{3}}{1+x^{4}}\right)dx
is equal to
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2\pi
0%
\dfrac{\pi}{2}
0%
0
0%
\pi
\displaystyle \int_{0}^{\pi /2}\sin x\log \left ( \sin x \right )dx=
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\displaystyle \log _{e}e
0%
\displaystyle \log _{e}2
0%
\displaystyle \log _{e}\left ( e/2 \right )
0%
\displaystyle \log _{e}\left ( 2/e \right )
Explanation
\displaystyle I= \int_{0}^{\pi /2}\sin x\log \left ( \sin x \right )dx
\displaystyle = \frac{1}{2}\int_{0}^{\pi /2}\sin x\log \sin ^{2}x dx
\displaystyle = \frac{1}{2}\int_{0}^{\pi /2}\sin x\log \left ( 1-\cos ^{2}x \right )dx
\displaystyle = \frac{1}{2}\int_{0}^{1}\log \left ( 1-t^{2} \right )dt
where
\displaystyle t= \cos x
\displaystyle = \frac{1}{2}\int_{0}^{1}\left [ -t^{2}-\frac{\left ( t^{2} \right )^{2}}{2}-\frac{\left ( t^{2} \right )^{3}}{3}-\cdots \right ]dt
\displaystyle = -\left [ \frac{1}{2.3}+\frac{1}{4.5}+\frac{1}{6.7}+\cdots \right ]
\displaystyle = -\left [ \left ( \frac{1}{2}-\frac{1}{3} \right )+\left ( \frac{1}{4}-\frac{1}{5} \right )+\left ( \frac{1}{6}-\frac{1}{7} \right )+\cdots \right ]
\displaystyle = -1+\left ( 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots \right )
.
\displaystyle = -\log _{e}e+\log _{e}\left ( 1+1 \right )= \log _{e}\left ( 2/e \right )
\displaystyle \int_{0}^{\infty}\frac{1}{1+x^{n}}dx,\:\forall\:n\:> 1
is equal to?
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\displaystyle2 \int_{0}^{\infty}\frac{1}{1+x}dx
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\displaystyle \int_{-\infty}^{\infty}\frac{1}{1+x^{n}}dx
0%
\displaystyle \int_{1}^{\infty}\frac{dx}{(x^{n}-1)^{1/n}}
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\displaystyle \int_{0}^{1}\frac{1}{(1-x^{n})^{1/n}}
Explanation
Let
\displaystyle I=\int _{ 0 }^{ \infty } \frac { 1 }{ 1+x^{ n } } dx=\int _{ \infty }^{ 0 } \frac { -dx }{ 1+x^{ n } }
Substitute
\displaystyle x=\frac { 1 }{ y } \Rightarrow dx=-\frac { 1 }{ { y }^{ 2 } }
\displaystyle I=\int _{ 0 }^{ \infty } \frac { 1 }{ y^{ 2 } } \frac { dy }{ 1+\left( \frac { 1 }{ y } \right) ^{ n } } =\int _{ 0 }^{ \infty } \frac { y^{ n-1 } }{ y(1+y^{ n }) } dy
Substitute
t^{ n }-1+y^{ n }\Rightarrow n{ t }^{ n-1 }dt=n{ y }^{ n-1 }dy
\displaystyle I=\int _{ 1 }^{ \infty } \frac { dt }{ t(t^{ n }-t)^{ 1/n } } =\int _{ \infty }^{ 1 } -\frac { 1 }{ t^{ 2 } } .\frac { dt }{ \left( 1-\frac { 1 }{ t^{ n } } \right) ^{ 1/n } }
Substitute
\displaystyle z=\frac { 1 }{ t } \Rightarrow dx=-\frac { 1 }{ { t }^{ 2 } } dt
\displaystyle I=\int _{ 0 }^{ 1 } \frac { dz }{ (1-z^{ n })^{ 1/n } } =\int _{ 0 }^{ 1 } \frac { dx }{ (1-x^{ n })^{ 1/n } }
If
\displaystyle I_1=\int_{0}^{\pi /2}\frac{x}{\sin x}dx
and
\displaystyle I_2=\int_{0}^{\pi /2}\frac{\tan ^{-1}x}{x}dx,
then
\displaystyle \frac{I_{1}}{I_{2}}=
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\displaystyle \frac{1}{2}
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1
0%
2
0%
\displaystyle \frac{\pi }{2}
Explanation
Substitute
\displaystyle x=\tan \theta \therefore I_2=\int_{0}^{\pi /4}\frac{\theta }{\tan \theta }.\sec ^{2}\theta d\theta
or
\displaystyle I_{2}=\int_{0}^{\pi /4}\frac{2\theta }{2\sin \theta \cos \theta }d\theta =\int_{0}^{\pi /4} \frac{2\theta }{\sin 2\theta }d\theta
Now substitute
\displaystyle 2\theta =t \therefore I_{2}=\frac{1}{2}\int_{0}^{\pi /2}\frac{t}{\sin t}dt=\frac{1}{2}I_{1}
\displaystyle \therefore \frac{I_{1}}{I_{2}}=2\Rightarrow \left ( C \right )
\displaystyle If \int_{-1}^{1}\frac{g\left ( x \right )}{1+t^{2}}dt= f\left ( x \right ) , where, g\left ( x \right )= \sin x
, then
{f}'\left ( \frac{\pi }{3} \right )
equals
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0%
\displaystyle\frac{\pi }{4}
0%
does not exist
0%
\displaystyle \frac{\pi \sqrt{3}}{4}
0%
None of these
Explanation
\displaystyle f\left( x \right) =\int _{ -1 }^{ 1 }{ \frac { \sin { x } }{ 1+{ t }^{ 2 } } } dt=\sin { x } \int _{ -1 }^{ 1 }{ \frac { 1 }{ 1+{ t }^{ 2 } } } dt
\displaystyle =\sin { x } \left[ \tan ^{ -1 }{ t } \right] _{ -1 }^{ 1 }=\sin { x } \left[ \frac { \pi }{ 4 } +\frac { \pi }{ 4 } \right] =\sin { x } \left[ \frac { \pi }{ 2 } \right]
\displaystyle \therefore f'\left( x \right) =\frac { \pi }{ 2 } \cos { x } \Rightarrow f'\left( \frac { \pi }{ 3 } \right) =\frac { \pi }{ 2 } \frac { 1 }{ 2 } =\frac { \pi }{ 4 }
The value of
\displaystyle \int_{1}^{1/e}f(x)dx+\int_{1}^{e}f(x)dx
where
f(x)
is given as
\displaystyle \frac{log\:x}{1+x}
equals
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0%
0
0%
\displaystyle -\frac{1}{2}
0%
\displaystyle \frac{1}{2}
0%
1
Explanation
Let
\displaystyle l_{ 1 }=\int _{ 1 }^{ 1/e } \frac { logx }{ 1+x } dx
and
\displaystyle l_{ 2 }=\int _{ 1 }^{ 0 } \frac { log\: x }{ 1+x } dx
Now for
\displaystyle l_{ 1 }=\int _{ 1 }^{ 1/e } \frac { log\: t }{ 1+x }
Substitute
\displaystyle x=\frac { 1 }{ t } \Rightarrow dx=-\frac { 1 }{ { t }^{ 2 } }
l_{ 1 }=\int _{ 1 }^{ e } \dfrac { log\: t }{ t(1+t) } dt=\int _{ 1 }^{ e } \dfrac { log\: x }{ x(1+x) } dx
( Replacing
t\rightarrow \: x
)
\displaystyle \therefore l_{ 1 }+l_{ 2 }=\int _{ 1 }^{ e } \frac { log\: x }{ x(1+x) } dx+\int _{ 1 }^{ e } \frac { log\: x }{ 1+x } dx
\displaystyle =\int _{ 1 }^{ e } \frac { (log(x))(1+x) }{ x(1+x) } dx=\int _{ 1 }^{ e } \frac { log\: x }{ x } dx
\displaystyle =\frac { 1 }{ 2 } \left[ (log\: x) \right] _{ 1 }^{ e }=\frac { 1 }{ 2 } [log_{ e }e-log_{ e }1]=\frac { 1 }{ 2 }
The value of
\displaystyle \int_{1/e}^{\tan x}\displaystyle \frac{t}{1+t^{2}}\, dt\, +\, \displaystyle \int_{1/e}^{\cot x}\displaystyle \frac{dt}{t\left ( 1+t^{2} \right )}
is
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0%
1/2
0%
1
0%
\pi /4
0%
none of these
Explanation
\displaystyle \int _{ 1/e }^{ \tan x } \frac { t }{ 1+t^{ 2 } } \, dt\, +\, \int _{ 1/e }^{ \cot x } \frac { dt }{ t\left( 1+t^{ 2 } \right) }
\displaystyle =\int _{ 1/e }^{ \tan x } \frac { t }{ 1+t^{ 2 } } \, dt\, +\, \int _{ 1/e }^{ \cot x } \left( \frac { 1 }{ t } -\frac { 1 }{ 1+{ t }^{ 2 } } \right) dt
=\dfrac { 1 }{ 2 } { \left[ \log { \left( 1+{ t }^{ 2 } \right) } \right] }_{ 1/e }^{ \tan { x } }+{ \left[ \log { t } \right] }_{ 1/e }^{ \cot { x } }-{ \left[ \tan ^{ -1 }{ t } \right] }_{ 1/e }^{ \cot { x } } =1
Evaluate :
\displaystyle \int_{-\frac{1}{\sqrt2}}^{\frac{1}{\sqrt2}}\frac{x^{8}}{1-x^{4}}\times \left [ \sin ^{-1}\left ( 1-2x^{2} \right ) +\cos ^{-1}\left ( 2x\sqrt{1-x^{2}} \right )\right ]dx
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0%
\displaystyle \pi \left [ \frac{1}{2}\log \frac{\sqrt{2}+1}{\sqrt{2}-1}+\tan ^{-1} \frac{1}{\sqrt{2}}-\frac{21}{10\sqrt{2}}\right ]
0%
\displaystyle \pi \left [ \frac{1}{2}\log \frac{\sqrt{2}+1}{\sqrt{2}-1}+\tan ^{-1} \frac{1}{\sqrt{2}}+\frac{21}{10\sqrt{2}}\right ]
0%
\displaystyle \pi \left [ \frac{1}{2}\log \frac{\sqrt{2}-1}{\sqrt{2}-1}+\tan ^{-1} \frac{1}{\sqrt{2}}-\frac{21}{10\sqrt{2}}\right ]
0%
\displaystyle \pi \left [ \frac{1}{2}\log \frac{\sqrt{2}-1}{\sqrt{2}-1}+\tan ^{-1} \frac{1}{\sqrt{2}}+\frac{21}{10\sqrt{2}}\right ]
Explanation
In
\displaystyle N^{r}
put
\displaystyle x=\sin \theta ,
then
\displaystyle \sin ^{-1}\left ( 1-2\sin ^{2}\theta \right )+\cos ^{-1}\left ( 2\sin \theta \cos \theta \right )=\sin ^{-1}\left ( \cos 2\theta \right )+\cos ^{-1}\left ( \sin 2\theta \right )
\displaystyle =\frac{\pi }{2}-\cos ^{-1}\left ( \cos 2\theta \right )+\frac{\pi }{2}-\sin ^{-1}\left ( \sin 2\theta \right )
\displaystyle =\pi -2\theta -2\theta =\pi -4\sin ^{-1}x
\displaystyle \therefore I=\int_{-\frac{1}{\sqrt2}}^{\frac{1}{\sqrt2}}\frac{x^{8}}{1-x^{4}}\left [ \pi -4\sin -\mid x \right ]dx
\displaystyle =2 \int_{0}^{\frac{1}{\sqrt{2}}}\pi \frac{x^{8}}{1-x^{4}}dx+0
by Prop.V
\displaystyle =2\pi \int_{0}^{\frac{1}{\sqrt{2}}}\frac{x^{8}-1+1}{1-x^{4}}dx
\displaystyle =2\pi \int_{0}^{\frac{1}{\sqrt{2}}}\left [ -\left ( x^{4}+1 \right )+\frac{1}{\left ( 1-x^{2} \right )\left ( 1+x^{2} \right )} \right ]dx
\displaystyle =2\pi \int_{0}^{\frac{1}{\sqrt{2}}}-\left ( x^{4}+1 \right )+\frac{1}{2}\left \{ \frac{1}{1-x^{2}}+\frac{1}{1+x^{2}} \right \}dx
\displaystyle =2x\left [ -\left ( \frac{x^{5}}{5}+x \right )+\frac{1}{4}\log \frac{1+x}{1-x} +\frac{1}{2}\tan -1x\right ]^{1/\sqrt{2}}_{0}
\displaystyle =\pi \left [ \frac{1}{2}\log \frac{\sqrt{2}+1}{\sqrt{2}-1}+\tan ^{-1} \frac{1}{\sqrt{2}}-\frac{21}{10\sqrt{2}}\right ]
Ans: A
Solve :-
\int_0^1 {\frac{{dx}}{{{{({x^2} + 1)}^{3/2}}}} = }
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0%
1
0%
\frac{1}{{\sqrt 2 }}
0%
\frac{1}{2}
0%
\sqrt 2
Solve:-
\int\limits_0^1 {\frac{{dx}}{{{{({x^2} + 1)}^{3/2}}}}}
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0%
1
0%
\sqrt 2
0%
1/2
0%
\dfrac{1}{\sqrt 2}
If
I= \displaystyle \int_{0}^{1/\sqrt{3}}\displaystyle \frac{dx}{\left ( 1+x^{2} \right )\sqrt{1-x^{2}}}
then
I
is equal to
Report Question
0%
\pi /2
0%
\pi /2\sqrt{2}
0%
\pi /4\sqrt{2}
0%
\pi /4
Explanation
Let
\displaystyle I=\int { \frac { 1 }{ \left( 1+{ x }^{ 2 } \right) \sqrt { 1-{ x }^{ 2 } } } dx }
Substitute
x=\sin { t } \Rightarrow dx=\cos { t } dt
\displaystyle I=\int { \frac { 1 }{ \sin ^{ 2 }{ t } +1 } dt }
Multiply numerator and denominator by
\csc ^{ 2 }{ t }
\displaystyle I=\int { \frac { \csc ^{ 2 }{ t } }{ \csc ^{ 2 }{ t } +1 } dt } =\int { \frac { \csc ^{ 2 }{ t } }{ \cot ^{ 2 }{ t } +2 } dt }
Substitute
u=\cot { t } \Rightarrow du=-\csc ^{ 2 }{ t } dt
\displaystyle I=-\int { \frac { 1 }{ { u }^{ 2 }+2 } du } =-\frac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \frac { u }{ \sqrt { 2 } } }
\displaystyle =-\frac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \frac { \cot { t } }{ \sqrt { 2 } } } =-\frac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \frac { \cot { \sin ^{ -1 }{ x } } }{ \sqrt { 2 } } }
Therefore
\displaystyle \int _{ 0 }^{ 1/\sqrt { 3 } }{ \frac { 1 }{ \left( 1+{ x }^{ 2 } \right) \sqrt { 1-{ x }^{ 2 } } } dx } =\frac { \pi }{ 4\sqrt { 2 } }
\displaystyle \int_{0}^{\pi /3}\frac{\cos \theta }{3+4\sin \theta }d\theta =\lambda \log \frac{3+2\sqrt{3}}{3}
then
\displaystyle \lambda
equals
Report Question
0%
\dfrac{1}{2}
0%
\dfrac{1}{3}
0%
\dfrac{1}{4}
0%
\dfrac{1}{8}
Explanation
Given
\displaystyle \int_{0}^{\pi /3}\frac{\cos \theta }{3+4\sin \theta }d\theta =\lambda \log \frac{3+2\sqrt{3}}{3}
....(1)
Consider,
I= \int _{ 0 }^{ \pi /3 } \dfrac { \cos \theta }{ 3+4\sin \theta } d\theta
Substitute
\sin \theta =t
\Rightarrow \cos \theta d\theta =dt
I=\displaystyle \int _{ 0 }^{ \sqrt { 3 } /2 } \frac { dt }{ 3+4t }
Substitute
3+4t=u
4dt=du
\therefore I=\displaystyle \frac {1}{4}\int _{ 3 }^{ 3+2\sqrt { 3 } } \frac { du }{ u }
\displaystyle =\frac{1}{4}[\log { u } ]_{ 3 }^{3+ 2\sqrt { 3 } }
\displaystyle I=\frac{1}{4} [\log { (3+2\sqrt { 3 } )-\log { 3 } }]
\Rightarrow I=\displaystyle \frac{1}{4}\log \frac { 3+2\sqrt { 3 } }{ 3 }
So, on comparing with (1), we get
\lambda =\displaystyle \frac{1}{4}
The value of
\displaystyle \int ^{\tan x}_{1/e}\displaystyle \frac{t\, dt}{1+t^{2}}+\displaystyle \int ^{\cot x}_{1/e}\displaystyle \frac{dt}{t\left ( 1+t^{2} \right )}
is
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\displaystyle \frac{1}{2+tan^{2}x}
0%
1
0%
\pi /4
0%
\displaystyle \frac{2}{\pi }\displaystyle \int ^{1}_{-1}\displaystyle \frac{dt}{1+t^{2}}
Explanation
\displaystyle \int _{ 1/e }^{ \tan x } \frac { t }{ 1+t^{ 2 } } \, dt\, +\, \int _{ 1/e }^{ \cot x } \frac { dt }{ t\left( 1+t^{ 2 } \right) }
\displaystyle =\int _{ 1/e }^{ \tan x } \frac { t }{ 1+t^{ 2 } } \, dt\, +\, \int _{ 1/e }^{ \cot x } \left( \frac { 1 }{ t } -\frac { 1 }{ 1+{ t }^{ 2 } } \right) dt
=\frac { 1 }{ 2 } { \left[ \log { \left( 1+{ t }^{ 2 } \right) } \right] }_{ 1/e }^{ \tan { x } }+{ \left[ \log { t } \right] }_{ 1/e }^{ \cot { x } }-{ \left[ \tan ^{ -1 }{ t } \right] }_{ 1/e }^{ \cot { x } }\\ =1
If
\displaystyle I_{n}= \int_{0}^{\pi /4}\tan ^{n}x\times \sec ^{2}x dx,
then
\displaystyle I_{1}, I_{2}, I_{3},
.......are in
Report Question
0%
A.P.
0%
G.P.
0%
H.P.
0%
none
Explanation
I_{ n }=\int _{ 0 }^{ \pi /4 } \tan ^{ n }{ x } \sec ^{ 2 } xdx
Let,
t=\tan { x }
\Rightarrow \sec ^{ 2 } xdx=dt
Therefore,
I_{ n }=\int _{ 0 }^{ 1 }{ { t }^{ n } } dt={ \left( \dfrac { t }{ n+1 } \right) }_{ 0 }^{ 1 }=\frac { 1 }{ n }
Therefore,
I_{ 1 },I_{ 2 },I_{ 3 }\cdots
are
\dfrac { 1 }{ 2 } ,\dfrac { 1 }{ 3 } ,\dfrac { 1 }{ 4 } ,\cdots
which are in H.P
Ans: C
The value of the integral
\displaystyle \int_{\alpha }^{\beta }\displaystyle \dfrac{dx}{\sqrt{\left ( x-\alpha \right )\left ( \beta -x \right )}}
for
\beta > \alpha
, is
Report Question
0%
\sin ^{-1}\: \alpha /\beta
0%
\pi /2
0%
\sin ^{-1}\beta /2\alpha
0%
\pi
Explanation
Given :
\displaystyle \int _{ \alpha }^{ \beta }{ \cfrac { dx }{ \sqrt { (x-\alpha )(\beta -x) } } }
\displaystyle I=\int _{ \alpha }^{ \beta }{ \cfrac { dx }{ \sqrt { (x-\alpha )(\beta -x) } } } =\int _{ \alpha }^{ \beta }{ \cfrac { dx }{ \sqrt { -\alpha \beta +x(\beta +\alpha )-{ x }^{ 2 } } } }
\displaystyle I=\int _{ \alpha }^{ \beta }{ \cfrac { dx }{ \sqrt { \cfrac { { (\alpha +\beta ) }^{ 2 } }{ 4 } -\alpha \beta -\left[ x-\cfrac { \beta +\alpha }{ 2 } \right] ^{ 2 } } } }
\displaystyle I=\int _{ \alpha }^{ \beta }{ \cfrac { dx }{ \sqrt { \cfrac { (\beta -\alpha )^{ 2 } }{ 4 } -\left[ x-\cfrac { (\beta +\alpha ) }{ 2 } \right] ^{ 2 } } } }
we know that,
\displaystyle \int \dfrac{1}{\sqrt{a^2-x^2}}dx =\sin^{-1}\left(\dfrac {x}{a}\right)+c
I=\sin ^{ -1 }{ \left[ \cfrac { x-\left( \cfrac { \beta +\alpha }{ 2 } \right) }{ \left( \cfrac { \beta -\alpha }{ 2 } \right) } \right] } _{ \alpha }^{ \beta }=\sin ^{ -1 }{ \left[ \cfrac { \left( \cfrac { \beta -\alpha }{ 2 } \right) }{ \left( \cfrac { \beta +\alpha }{ 2 } \right) } \right] } -\sin ^{ -1 }{ \left[ \cfrac { \left( \cfrac { \alpha -\beta }{ 2 } \right) }{ \left( \cfrac { \beta -\alpha }{ 2 } \right) } \right] }
I=\sin ^{ -1 }{ (1)-\sin ^{ -1 }{ (-1) } }
I=\cfrac { \pi }{ 2 } +\cfrac { \pi }{ 2 } =\pi
Hence the correct answer is
\pi
\displaystyle \int_{\pi /6}^{\pi /4}\frac{dx}{\sin 2x}
is equal to
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0%
\displaystyle \frac{1}{2}\log \left ( -1 \right )
0%
\displaystyle \log \left ( -1 \right )
0%
\displaystyle \log 3
0%
\displaystyle \frac {1}{2} \log \sqrt{3}
Explanation
Let
\displaystyle I=\int { \frac { 1 }{ \sin { 2x } } dx } =\int { \csc { 2x } dx }
Put
2x=t\Rightarrow 2dx=dt
\displaystyle I=\frac { 1 }{ 2 } \int { \csc { t } dt }
Multiply numerator and denominator by
\cot { t } +\csc { t }
\displaystyle I=\frac { 1 }{ 2 } \int { -\frac { -\csc ^{ 2 }{ t } -\cot { t } \csc { t } }{ \cot { t } +\csc { t } } dt }
Put
\cot { t } +\csc { t } =u\Rightarrow \left( -\csc ^{ 2 }{ t } -\cot { t } \csc { t } \right) dt=du
\displaystyle I=-\frac { 1 }{ 2 } \int { \frac { 1 }{ u } du } =-\frac { 1 }{ 2 } \log { u } =-\frac { 1 }{ 2 } \log { \left( \cot { t } +\csc { t } \right) }
\displaystyle =-\frac { 1 }{ 2 } \log { \left( \cot { 2x } +\csc { 2x } \right) } =-\frac { 1 }{ 2 } \log { \left( \cot { x } \right) }
Hence
\displaystyle \int _{ \pi /6 }^{ \pi /4 }{ \frac { 1 }{ \sin { 2x } } dx } =-\frac { 1 }{ 2 } { \left[ \log { \left( \cot { x } \right) } \right] }_{ \pi /6 }^{ \pi /4 }=\frac { 1 }{ 2 } \log { \sqrt { 3 } }
Value of
\displaystyle \int_{0}^{1}\displaystyle \frac{dx}{\left ( 1+x^{2} \right )\sqrt{1-x^{2}}}
is?
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0%
\dfrac{\pi}{2\sqrt{2}}
0%
\dfrac{\pi}{\sqrt{2}}
0%
\sqrt{2}\pi
0%
2\sqrt{2\pi }
Explanation
Let
\displaystyle I=\int { \frac { dx }{ \left( 1+x^{ 2 } \right) \sqrt { 1-x^{ 2 } } } }
Substitute
\displaystyle x=\frac { 1 }{ t } \Rightarrow dx=-\frac { 1 }{ { t }^{ 2 } } dt
\displaystyle I=\int { \frac { -tdt }{ \left( { t }^{ 2 }+1 \right) \sqrt { { t }^{ 2 }-1 } } }
Substitute
{ t }^{ 2 }-1={ u }^{ 2 }\Rightarrow 2tdt=2udu
\displaystyle I=-\int { \frac { udu }{ \left( { u }^{ 2 }+1 \right) u } } =-\int { \frac { 1 }{ { u }^{ 2 }+{ \left( \sqrt { 2 } \right) }^{ 2 } } du }
\displaystyle =-\frac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \left( \frac { u }{ \sqrt { 2 } } \right) } =-\frac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \left( \frac { \sqrt { 2 } x }{ \sqrt { 1-{ x }^{ 2 } } } \right) }
Therefore
\displaystyle \int _{ 0 }^{ 1 }{ Idx } =\frac { \pi }{ 2\sqrt { 2 } }
If
3+2\displaystyle\int _{ 0 }^{ 1 }{ { x }^{ 2 }{ e }^{ -x^{ 2 } } } dx=\displaystyle\int _{ 0 }^{ 1 }{ { e }^{ -x^{ 2 } } } dx
then the value of
\beta
is
Report Question
0%
\dfrac{1}{e}
0%
e
0%
\dfrac{1}{2e}
0%
2e
If
I_n=\displaystyle\int^1_0\dfrac{dx}{(1+x^2)^n}; n\in N
, then which of the following statements hold good?
Report Question
0%
2n I_{n+1}=2^{-n}+(2n-1)I_n
0%
I_2=\dfrac{\pi}{8}+\dfrac{1}{4}
0%
I_2=\dfrac{\pi}{8}-\dfrac{1}{4}
0%
I_3=\dfrac{\pi}{16}-\dfrac{5}{48}
If
\int { \frac { { x }^{ 1/2 } }{ { x }^{ 1/2 }-{ x }^{ 1/3 } } } dx=Ax\frac { 6 }{ 5 } { x }^{ 5/6 }+\frac { 3 }{ 2 } { x }^{ 2/3 }+B\sqrt { x } +{ Cx }^{ 1/3 }+{ Dx }^{ 1/6 }+E
In
\left( { x }^{ 1/6 }-1 \right) =k,
then A+B+C+D+E=
Report Question
0%
6
0%
12
0%
18
0%
17
The value of
\displaystyle \int_{1/2}^{1}\displaystyle \frac{dx}{x\sqrt{3x^{2}+2x-1}}
is?
Report Question
0%
\pi /2
0%
\pi /3
0%
\pi /6
0%
\pi /\sqrt{2}
Explanation
Let
\displaystyle I=\int { \frac { 1 }{ x\sqrt { { 3x }^{ 2 }+2x-1 } } } dx=\int { \frac { 1 }{ x\sqrt { { \left( \sqrt { 3 } x+\frac { 1 }{ \sqrt { 3 } } \right) }^{ 2 }-\frac { 4 }{ 3 } } } } dx
Put
\displaystyle t=\sqrt { 3 } x+\frac { 1 }{ \sqrt { 3 } } \Rightarrow dt=\sqrt { 3 } dt
\displaystyle I=\frac { 1 }{ \sqrt { 3 } } \int { -\frac { 1 }{ \left( \sqrt { 3 } -3t \right) \sqrt { { t }^{ 2 }-\frac { 4 }{ 3 } } } } dt
\displaystyle =-3\int { \frac { 1 }{ \left( \sqrt { 3 } -3t \right) \sqrt { { t }^{ 2 }-\frac { 4 }{ 3 } } } } dt
\displaystyle I=3\int { \frac { 2 }{ 3\sqrt { 3 } { s }^{ 2 }+\sqrt { 3 } } ds } =2\sqrt { 3 } \int { \frac { 1 }{ 3{ s }^{ 2 }+1 } ds }
\displaystyle =2\tan ^{ -1 }{ \left( \sqrt { 3 } s \right) } =\tan ^{ -1 }{ \left( \frac { x-1 }{ \sqrt { 3{ x }^{ 2 }+2x-1 } } \right) }
\displaystyle \therefore \int _{ 1/2 }^{ 1 } { \frac { 1 }{ x\sqrt { { 3x }^{ 2 }+2x-1 } } } { dx } =\frac { \pi }{ 6 }
If
I= \displaystyle \int_{1}^{\infty }\displaystyle \frac{x^{2}-2}{x^{3}\sqrt{x^{2}-1}}\: dx
, then
I
equals
Report Question
0%
-1
0%
0
0%
\pi /2
0%
\pi -\sqrt{3}
Explanation
\displaystyle I=\int _{ 1 }^{ \infty }{ \frac { { x }^{ 2 }-2 }{ { x }^{ 3 }\sqrt { { x }^{ 2 }-1 } } dx }
Put Put
x=\sec { u } \Rightarrow dx=\tan { u } \sec { u } du
\displaystyle I=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \cos ^{ 2 }{ u\left( \sec ^{ 2 }{ u } -2 \right) } du } =\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 1-\sin ^{ 2 }{ u } \right) \left( \sec ^{ 2 }{ u } -2 \right) du }
\displaystyle =\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2\sin ^{ 2 }{ u } -\tan ^{ 2 }{ u } +\sec ^{ 2 }{ u } -2 \right) } du=-\int _{ 0 }^{ \frac { \pi }{ 2 } }{ du } +2\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sin ^{ 2 }{ u } du }
\displaystyle =-\int _{ 0 }^{ \frac { \pi }{ 2 } }{ du } -\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \cos { 2udu } } +\int _{ 0 }^{ \frac { \pi }{ 2 } }{ du } =-{ \left[ \frac { \sin { 2u } }{ 2 } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }=0
If
0< \alpha < \pi /2
then the value of
\displaystyle \int_{0}^{\alpha }\displaystyle \frac{dx}{1-\cos x\cos \alpha }
is
Report Question
0%
\pi /\alpha
0%
\pi /2\sin \alpha
0%
\pi /2\cos \alpha
0%
\pi /2\alpha
Explanation
Let
\displaystyle I=\int _{ 0 }^{ \alpha } \dfrac { dx }{ 1-\cos x\cos \alpha }
\displaystyle =\int _{ 0 }^{ \alpha }{ \dfrac { dx }{ \left( \cos ^{ 2 }{ \dfrac { x }{ 2 } } +\sin ^{ 2 }{ \dfrac { x }{ 2 } } \right) -\cos { \alpha } \left( \cos ^{ 2 }{ \dfrac { x }{ 2 } } -\sin ^{ 2 }{ \dfrac { x }{ 2 } } \right) } }
\displaystyle =\int _{ 0 }^{ \alpha }{ \dfrac { dx }{ \left( 1-\cos { \alpha } \right) \cos ^{ 2 }{ \dfrac { x }{ 2 } } +2\cos ^{ 2 }{ \dfrac { \alpha }{ 2 } } \sin ^{ 2 }{ \dfrac { x }{ 2 } } } }
\displaystyle =\dfrac { 1 }{ 2 } \int _{ 0 }^{ \alpha }{ \dfrac { \sec ^{ 2 }{ \dfrac { \alpha }{ 2 } } \sec ^{ 2 }{ \dfrac { x }{ 2 } } }{ \tan ^{ 2 }{ \dfrac { \alpha }{ 2 } } +\tan ^{ 2 }{ \dfrac { x }{ 2 } } } dx }
Substitute
\displaystyle \tan { \dfrac { x }{ 2 } } =t
\displaystyle I=\int _{ 0 }^{ \tan { \dfrac { \alpha }{ 2 } } }{ \dfrac { \sec ^{ 2 }{ \dfrac { \alpha }{ 2 } } }{ \tan ^{ 2 }{ \dfrac { \alpha }{ 2 } } +{ t }^{ 2 } } dt } =\sec ^{ 2 }{ \dfrac { \alpha }{ 2 } } \cot { \dfrac { \alpha }{ 2 } } { \left[ \tan ^{ -1 }{ \dfrac { 1 }{ \tan { \dfrac { \alpha }{ 2 } } } } \right] }_{ 0 }^{ \tan { \dfrac { \alpha }{ 2 } } }
\displaystyle =\dfrac { \pi }{ 2\sin { \alpha } }
Value of
\displaystyle \int_{a}^{\infty }\displaystyle \frac{dx}{x^{4}\sqrt{a^{2}+x^{2}}}
is
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0%
\displaystyle \frac{2+\sqrt{2}}{3a^{4}}
0%
\displaystyle \frac{2-\sqrt{2}}{3a^{2}}
0%
\displaystyle \frac{2-\sqrt{2}}{3a^{4}}
0%
\displaystyle \frac{\sqrt{2}+1}{3a^{2}}
Explanation
Let
\displaystyle I=\int _{ a }^{ \infty }{ \frac { 1 }{ { x }^{ 4 }\sqrt { { a }^{ 2 }+x^{ 2 } } } } dx
Substitute
x=a\tan { t\Rightarrow } dx=a\sec ^{ 2 }{ t } dt
\displaystyle I=a\int _{ \frac { \pi }{ 4 } }^{ \frac { \pi }{ 2 } }{ \frac { \cot ^{ 3 }{ t } \csc { t } }{ { a }^{ 5 } } } dt
\displaystyle =\frac { 1 }{ { a }^{ 4 } } \int _{ \frac { \pi }{ 4 } }^{ \frac { \pi }{ 2 } }{ \cot { t } \csc { t } \left( \csc ^{ 2 }{ t-1 } \right) dt }
Substitute
\upsilon =\csc { t } \Rightarrow d\upsilon =-\cot { t } \csc { t } dt
\displaystyle I=\frac { 1 }{ { a }^{ 4 } } \int _{ \sqrt { 2 } }^{ 1 }{ \left( { \upsilon }^{ 2 }-1 \right) } d\upsilon
\displaystyle =\frac { 1 }{ { a }^{ 2 } } \left[ \frac { { \upsilon }^{ 2 } }{ 3 } -\upsilon \right] _{ \sqrt { 2 } }^{ 1 }{ =\frac { 2-\sqrt { 2 } }{ { 3a }^{ 4 } } }
The value of
\displaystyle \int_{-4}^{-5}e^{\left ( x+5 \right )^{2}}dx+3\displaystyle \int_{1/3}^{2/3}e^{9\left ( x-2/3 \right )^{2}}dx
is
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0%
2/5
0%
1/5
0%
1/2
0%
none of these
Explanation
Let
I=\int _{ -4 }^{ -5 } e^{ \left( x+5 \right) ^{ 2 } }dx+3\int _{ 1/3 }^{ 2/3 } e^{ 9\left( x-2/3 \right) ^{ 2 } }dx={ I }_{ 1 }+{ I }_{ 2 }
Where
{ I }_{ 1 }=\int _{ -4 }^{ -5 } e^{ \left( x+5 \right) ^{ 2 } }dx
Put
x+5=t
{ I }_{ 1 }=-\int _{ 0 }^{ 1 }{ { e }^{ { t }^{ 2 } } } dt
And
{ I }_{ 2 }=3\int _{ 1/3 }^{ 2/3 } e^{ 9\left( x-2/3 \right) ^{ 2 } }dx
Put
-3x+2=u
{ I }_{ 2 }=\int _{ 0 }^{ 1 }{ { e }^{ { u }^{ 2 } } } du
\therefore I=0
Value of
\displaystyle \int_{0}^{16}\displaystyle \frac{x^{1/4}}{1+x^{1/2}}\: dx
is
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\displaystyle \frac{8}{3}
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\displaystyle \frac{4}{3}\tan ^{-1}\: 2
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4\displaystyle \left ( \displaystyle \frac{2}{3}+\tan ^{-1}\: 2 \right )
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4\displaystyle \left ( \displaystyle \frac{2}{3}-\tan ^{-1}\: 2 \right )
Explanation
\displaystyle I=\int _{ 0 }^{ 16 }{ \frac { \sqrt [ 4 ]{ x } }{ 1+\sqrt { x } } dx }
Put
\displaystyle u=\sqrt [ 4 ]{ x } \Rightarrow du=\frac { 1 }{ 4{ x }^{ \frac { 3 }{ 4 } } } dx
\displaystyle I=4\int _{ 0 }^{ 2 }{ \left( { u }^{ 2 }+\frac { 1 }{ { u }^{ 2 }+1 } -1 \right) } du
\displaystyle =4\int _{ 0 }^{ 2 }{ { u }^{ 2 } } du+4\int _{ 0 }^{ 2 }{ \frac { 1 }{ { u }^{ 2 }+1 } du } -4\int _{ 0 }^{ 2 }{ du }
\displaystyle =4\left[ \frac { { u }^{ 3 } }{ 3 } \right] _{ 0 }^{ 2 }{ + }4\left[ \tan ^{ -1 }{ u } \right] _{ 0 }^{ 2 }-4\left[ u \right] _{ 0 }^{ 2 }
\displaystyle =4\left( \frac { 2 }{ 3 } +\tan ^{ -1 }{ 2 } \right)
The solution of
x
of the equation
\displaystyle \int_{\sqrt{2}}^{x}{\dfrac{dt}{t\sqrt{t^{2}-1}}}=\dfrac{\pi}{2}
is
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2\sqrt{2}
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2
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\pi
0%
-\sqrt{2}
\int_{0}^{1}\frac{1}{ax+b}dx is
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zero
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log_{e}\frac{a+b}{b}
0%
log_{e}(ax+b)
0%
\frac{1}{a}log_{e}\left ( \frac{a+b}{b} \right )
\int _{ 0 }^{ 8 }{ \left[ \sqrt { t } \right] } dt
at equals to (where [.] greatest integer function.)
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0%
28
0%
11
0%
2
0%
8
If
\displaystyle \frac{\pi }{4}< \alpha < \displaystyle \frac{\pi }{2}
, value of
\displaystyle \int_{-\pi /2}^{\pi /2}\displaystyle \frac{\sin 2x}{\sqrt{1+\sin 2\alpha \sin x}}
is
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-\displaystyle \frac{4}{3}\tan \alpha \sec \alpha
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-\displaystyle \frac{4}{3}\cot \alpha cosec\alpha
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-\displaystyle \frac{4}{3}\tan \alpha cosec\alpha
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-\displaystyle \frac{4}{3}\cot \alpha \sec \alpha
Explanation
Let
\displaystyle I=\int _{ -\pi /2 }^{ \pi /2 } \frac { \sin 2x }{ \sqrt { 1+\sin 2\alpha \sin x } } =2\int _{ -\pi /2 }^{ \pi /2 } \frac { \sin { x } \cos { x } }{ \sqrt { 1+\sin 2\alpha \sin x } } dx
Substitute
t=\sin { x } \Rightarrow dt=\cos { x } dx
\displaystyle I=2\int _{ -1 }^{ 1 }{ \frac { t }{ \sqrt { t\sin { 2\alpha } +1 } } dt }
Substitute
u=t\sin { 2\alpha } \Rightarrow du=\sin { 2\alpha } dt
\displaystyle I=\int _{ 1-\sin { 2\alpha } }^{ 1+\sin { 2\alpha } }{ 2\csc ^{ 2 }{ 2\alpha } \frac { u-1 }{ \sqrt { u } } }
\displaystyle { \left[ \frac { 4 }{ 3 } { u }^{ \frac { 3 }{ 2 } }\csc ^{ 2 }{ 2\alpha } -4\sqrt { 5 } \csc ^{ 2 }{ 2\alpha } \right] }_{ 1-\sin { 2\alpha } }^{ 1+\sin { 2\alpha } }
\displaystyle =-\frac { 4 }{ 3 } \cot { \alpha } \csc { \alpha }
If
I_{1}= \displaystyle \int_{x}^{1}\displaystyle \frac{dt}{1+t^{2}}
and
I_{2}= \displaystyle \int_{1}^{1/x}\displaystyle \frac{dt}{1+t^{2}}
for
x> 0
, then
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I_{1}= I_{2}
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I_{1}> I_{2}
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I_{2}> I_{1}
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I_{2}= \left ( \pi /2 \right )-\tan ^{-1}x
If
I_{1}= \displaystyle \int_{0}^{\infty }\displaystyle \frac{dx}{1+x^{4}}
and
I_{2}= \displaystyle \int_{0}^{\infty }\displaystyle \frac{x^{2}}{1+x^{4}}\: dx
, then
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I_{1}= I_{2}
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I_{1}=2 I_{2}
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2I_{1}= I_{2}
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none of these
Explanation
I_{2}=\displaystyle \int _{0}^{\infty} \dfrac{x^2}{1+x^4}d{x}=\int^{\infty}_{0}\dfrac{\dfrac{1}{x^2}}{1+\dfrac{1}{x^4}}d{x}
Put
t=\dfrac{1}{x}\implies d{t}=-\dfrac{d{x}}{x^2}
I_{2}=\displaystyle\int _{\infty}^{0} \dfrac{-d{t}}{1+t^4}=\int_{0}^{\infty} \dfrac{d{x}}{1+x^4}=I_{1}
If
\displaystyle \int_{0}^{\infty }\displaystyle \frac{\log \left ( 1+x^{2} \right )}{1+x^{2}}\: dx= \lambda \displaystyle \int_{0}^{1}\displaystyle \frac{\log \left ( 1+x \right )}{1+x^{2}}\: dx
then
\lambda
equals
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4
0%
\pi
0%
8
0%
2\pi
Explanation
Let
I_1=\displaystyle \int_{0}^{\infty }\displaystyle \frac{\log \left ( 1+x^{2} \right )}{1+x^{2}}\: dx
and
I_2= \displaystyle \int_{0}^{1}\displaystyle \frac{\log \left ( 1+x \right )}{1+x^{2}}\: dx
Put
x=\tan\theta
in both integral
\Rightarrow dx=\sec^2\theta d\theta
I_1=\displaystyle \int_{0}^{\frac{\pi}{2} }\displaystyle \frac{\log \left ( \sec^2\theta \right )}{\sec^2\theta}\cdot \sec^2\theta d\theta=-\int_0^{\frac{\pi}{2}}\log\cos^2\theta d\theta ......(1)
Also using
\left( \int_0^a f(x)dx=\int_0^af(a-x)dx \right)
I_1 =\displaystyle -\int_0^{\frac{\pi}{2}}\log \sin^2\theta d\theta ........(2)
Adding (1) and (2) we get
2I_1 = \displaystyle -2 \int_0^{\frac{\pi}{2}}\log (\sin\theta\cos\theta)d\theta
\Rightarrow\displaystyle I_1=-\int_0^{\frac{\pi}{2}}\log\frac{\sin2\theta}{2} d\theta =\log 2\int_0^{\frac{\pi}{2}}d\theta- \int_0^{\frac{\pi}{2}} \log(\sin2\theta)d\theta
Put
2\theta =x\Rightarrow 2d\theta=dx
\Rightarrow \displaystyle I_1= \frac{\pi}{2}\log 2 -\frac{1}{2}\int_0^{\pi}\log(\sin\theta)d\theta=\frac{\pi}{2}\log 2 -\int_0^{\frac{\pi}{2}}\log(\sin\theta)d\theta=\frac{\pi}{2}\log 2+\frac{I_1}{2}
, using (2)
\therefore \displaystyle I_1 =\pi\log 2
Now
I_2=\displaystyle \int_0^{\frac{\pi}{4}}\log(1+\tan\theta)d\theta
\Rightarrow \displaystyle I_2 =\int_0^{\frac{\pi}{4}}\log(1+\tan(\frac{\pi}{4}-\theta))d\theta=\int_0^{\frac{\pi}{4}}\log\left(1+\frac{1-\tan\theta}{1+\tan\theta}\right)d\theta
\displaystyle \Rightarrow I_2=\int_0^{\frac{\pi}{4}}\log\left(\frac{2}{1+\tan\theta}\right)d\theta=\log2\int_0^{\frac{\pi}{4}}d\theta-\displaystyle \int_0^{\frac{\pi}{4}}\log(1+\tan\theta)d\theta
\Rightarrow \displaystyle I_2=\frac{\pi}{4}\log 2-I_2 \Rightarrow I_2=\frac{\pi}{8}\log 2
Hence
\lambda = \cfrac{I_1}{I_2}=8
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers
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