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CBSE Questions for Class 12 Commerce Maths Integrals Quiz 4 - MCQExams.com
CBSE
Class 12 Commerce Maths
Integrals
Quiz 4
The value of
∫
π
0
d
x
1
−
2
α
cos
x
+
α
2
is
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0%
π
1
+
α
2
if
α
>
1
0%
π
α
2
−
1
if
α
>
1
0%
π
1
+
α
2
if
α
<
1
0%
π
α
2
−
1
if
α
<
1
Explanation
Let
I
=
∫
π
0
d
x
1
−
2
α
cos
x
+
α
2
=
∫
π
0
d
x
1
−
2
α
cos
x
+
α
2
Substitute
tan
x
2
=
t
⇒
1
2
sec
2
x
2
d
x
=
d
t
I
=
∫
∞
0
1
1
−
2
α
(
1
−
t
2
1
+
t
2
)
+
α
2
.
2
t
1
+
t
2
d
t
=
∫
∞
0
2
t
1
+
t
2
−
2
α
(
1
−
t
2
)
+
α
2
(
1
+
t
2
)
d
t
=
π
α
2
−
1
if
α
>
1
Given
∫
2
1
e
x
2
d
x
=
a
,
the value of
∫
e
4
e
√
ln
(
x
)
d
x
is?
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0%
e
4
−
e
0%
e
4
−
a
0%
2
e
4
−
a
0%
2
e
4
−
e
−
a
Explanation
Given
∫
2
1
e
x
2
d
x
=
a
,
Let
I
=
∫
e
4
e
√
ln
(
x
)
d
x
Put
ln
(
x
)
=
t
2
⇒
1
x
d
x
=
2
t
d
t
∴
The value of
\displaystyle \int_{0}^{1}\displaystyle \frac{dx}{\left ( x+1 \right )\sqrt{x^{2}+2x}}
is
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0%
\pi /6
0%
\pi /3
0%
\pi /2
0%
\pi
Explanation
Let
\displaystyle I=\int _{ 0 }^{ 1 }{ \frac { 1 }{ \left( x+1 \right) \sqrt { { x }^{ 2 }+2x } } dx } =\int _{ 0 }^{ 1 }{ \frac { 1 }{ \left( x+1 \right) \sqrt { { \left( x+1 \right) }^{ 2 }-1 } } dx }
Substitute
u=x+1\Rightarrow du=dx
\displaystyle\therefore I=\int _{ 1 }^{ 2 }{ \frac { 1 }{ u\sqrt { { u }^{ 2 }-1 } } du }
Substitute
\displaystyle s=\sqrt { { u }^{ 2 }-1 } \Rightarrow ds=\frac { u }{ \sqrt { { u }^{ 2 }-1 } } du
\displaystyle\therefore I=\int _{ 0 }^{ \sqrt { 3 } }{ \frac { 1 }{ { s }^{ 2 }+1 } ds } ={ \left[ \tan ^{ -1 }{ s } \right] }_{ 0 }^{ \sqrt { 3 } }=\frac { \pi }{ 3 }
Value of
\displaystyle \int_{0}^{\pi /4}\left ( \sqrt{\tan x}-\sqrt{\cot x} \right )\: dx
is
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0%
\sqrt{2}\log \left ( \sqrt{2}-1 \right )
0%
\sqrt{2}\log \left ( \sqrt{2}+1 \right )
0%
\log \left ( \sqrt{2}+1 \right )
0%
\log \left ( \sqrt{2}-1 \right )
Explanation
Let
\displaystyle I=\int _{ 0 }^{ \dfrac { \pi }{ 4 } }{ \left( \sqrt { \tan { x } } -\sqrt { cot{ x } } \right) } dx=-\int _{ 0 }^{ \dfrac { \pi }{ 4 } }{ \frac { cos{ x }-sin{ x } }{ \sqrt { cos{ x }sin{ x } } } dx }
Substitute
\left( sin{ x+cos{ x } } \right) =t\Rightarrow 2sin{ x }cos{ x }={ t }^{ 2 }-1
\displaystyle \therefore I=\sqrt { 2 } \int _{ 1 }^{ \sqrt { 2 } }{ \frac { dt }{ \sqrt { { t }^{ 2 }-1 } } } \\ =\left[ \sqrt { 2 } \log { \left( t+\sqrt { { t }^{ 2 } } -1 \right) } \right] _{ 0 }^{ \sqrt { 2 } }\\ =\sqrt { 2 } \log { \left( \sqrt { 2 } -1 \right) }
Value of
\displaystyle \int_{0}^{2a}\dfrac{x^{3/2}}{\sqrt{2a-x}} dx
is
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0%
\displaystyle \frac{3\pi a^{2}}{2}
0%
\pi a^{3}
0%
\sqrt{2}\pi a^{3}
0%
2\pi a^{3}
Explanation
Let
\displaystyle I=\int _{ 0 }^{ 2a }{ \frac { { x }^{ \frac { 3 }{ 2 } } }{ \sqrt { 2a-x } } dx }
Substitute
\displaystyle u=\sqrt { x } \Rightarrow du=\frac { 1 }{ 2\sqrt { x } } dx
\displaystyle I=2\int _{ 0 }^{ \sqrt { 2a } }{ \frac { { u }^{ 4 } }{ \sqrt { 2a-{ u }^{ 2 } } } du }
Substitute
u=\sqrt { 2a } \sin { t } \Rightarrow du=\sqrt { 2a } \cos { t } dt
\displaystyle I=2\sqrt { 2a } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2\sqrt { 2 } { a }^{ \frac { 3 }{ 2 } }\sin ^{ 4 }{ t } \right) dt } =8{ a }^{ 2 }\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \sin ^{ 4 }{ t } dt }
Using reduction formulae
\displaystyle \int { \sin ^{ m }{ x } dx } =\frac { -\cos { x } \sin ^{ m-1 }{ x } }{ m } +\frac { m-1 }{ m } \int { \sin ^{ m-2 }{ x } dx }
\displaystyle I={ \left[ -2{ a }^{ 2 }\sin ^{ 3 }{ t } \cos { t } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }+6{ a }^{ 2 }\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sin ^{ 2 }{ t } dt }
\displaystyle =0+6{ a }^{ 2 }\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( \frac { 1 }{ 2 } -\frac { 1 }{ 2 } \cos { 2t } \right) dt }
\displaystyle ={ \left[ 3{ a }^{ 2 }t-3{ a }^{ 2 }\sin { t } \cos { t } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }=\frac { 3\pi { a }^{ 2 } }{ 2 }
The value of
\displaystyle \int_{a}^{b}\displaystyle \frac{\log x}{x}\: dx
is
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0%
\log \left ( ab \right )\log \displaystyle \left ( \frac{b}{a} \right )
0%
\displaystyle \frac{1}{2}\log \left ( ab \right )\log \displaystyle \left ( \frac{b}{a} \right )
0%
\log \left ( a^{2}-b^{2} \right )
0%
\left ( a+b \right )\log \left ( a+b \right )
Explanation
\displaystyle I=\int _{ a }^{ b }{ \frac { \log { x } }{ x } } dx={ \left[ \log { x } .\log { x } \right] }_{ a }^{ b }-\int _{ a }^{ b }{ \frac { \log { x } }{ x } } dx\\ \Rightarrow 2I={ \left[ { \left( \log { x } \right) }^{ 2 } \right] }_{ a }^{ b }={ \left( \log { b } \right) }^{ 2 }-{ \left( \log { a } \right) }^{ 2 }
\displaystyle \Rightarrow I=\frac { 1 }{ 2 } \left( \log { b } +\log { a } \right) \left( \log { b } -\log { a } \right) =\frac { 1 }{ 2 } \log { ab } \log { \frac { b }{ a } }
Value of
\displaystyle \int_{0}^{25}\displaystyle \frac{1}{\sqrt{4+\sqrt{x}}}\: dx
is
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0%
2\left ( \sqrt{29}-1 \right )
0%
2\left ( \sqrt{29}-5 \right )
0%
3 \sqrt{29}-1
0%
none of these
Explanation
Let
\displaystyle I=\int _{ 0 }^{ 25 }{ \frac { 1 }{ \sqrt { \sqrt { x } +4 } } dx }
Substitute
\displaystyle u=\sqrt { x } \Rightarrow du=\frac { 1 }{ 2\sqrt { x } } dx
\displaystyle I=2\int _{ 0 }^{ 5 }{ \frac { u }{ \sqrt { u+4 } } du }
Substitute
s=u+4\Rightarrow ds=du
\displaystyle I=2\int _{ 4 }^{ 9 }{ \frac { s-4 }{ \sqrt { s } } ds } =2\int _{ 4 }^{ 9 }{ \left( \sqrt { s } -\frac { 4 }{ \sqrt { s } } \right) ds }
\displaystyle={ \left[ \frac { 4{ s }^{ 3/2 } }{ 3 } \right] }_{ 4 }^{ 9 }-{ \left[ 16\sqrt { s } \right] }_{ 4 }^{ 9 }=36-\frac { 32 }{ 3 } -48-32=-\frac { 164 }{ 3 }
\displaystyle \int_{0}^{\infty }f\left ( x+\frac{1}{x} \right )\frac{\ln x}{x}dx
Report Question
0%
is equal to zero
0%
is equal to one
0%
is equal to
\displaystyle \frac{1}{2}
0%
can not be evaluated
Explanation
I = \displaystyle \int_{0}^{\infty }f\left ( x+\frac{1}{x} \right )\frac{\ln x}{x}dx
Substitute
\displaystyle x = \frac{1}{t}\Rightarrow dx =-\frac{1}{t^2}dt
I = \displaystyle \int_{\infty }^0 f\left ( t+\frac{1}{t} \right )\frac{\ln (1/t)}{1/t} .\frac{-dt}{t^2}
\quad \displaystyle =\int_{\infty }^0 f\left ( t+\frac{1}{t} \right )\frac{\ln t}{t} dt
\quad \displaystyle =\int_{\infty }^0 f\left ( x+\frac{1}{x} \right )\frac{\ln x}{x} dx=-I
\Rightarrow 2I = 0\Rightarrow I = 0
If
\displaystyle \int_{0}^{\pi /3}\frac{\cos }{3+4\sin x}dx=K\log \frac{\left ( 3+2\sqrt{3} \right )}{3}
then K is
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0%
\displaystyle \frac{1}{2}
0%
\displaystyle \frac{1}{3}
0%
\displaystyle \frac{1}{4}
0%
\displaystyle \frac{1}{8}
Explanation
\displaystyle I=\int _{ 0 }^{ \pi /3 }{ \cfrac { \cos { x } }{ 3+4\sin { x } } dx }
Substituting
t=3+4\sin { x } \Rightarrow dt=4\cos { x }
\displaystyle I=\frac { 1 }{ 4 } \int _{ 3 }^{ 3+2\sqrt { 3 } }{ \frac { 1 }{ t } dt } ={ \left[ \frac { \log { t } }{ 4 } \right] }_{ 3 }^{ 3+2\sqrt { 3 } }
\displaystyle=\frac { \log { \left( 3+2\sqrt { 3 } \right) } -\log { 3 } }{ 4 } =\frac { 1 }{ 4 } \log { \left( \frac { 3+2\sqrt { 3 } }{ 3 } \right) } \Rightarrow K=\frac { 1 }{ 4 }
Suppose that F(x) is an antiderivative of f(x)
\displaystyle =\frac{\sin x}{x},x> 0
then
\displaystyle \int_{1}^{3}\frac{\sin 2x}{x}
can be expressed as
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0%
F(6) - F(2)
0%
\displaystyle \frac{1}{2}\left ( F\left ( 6 \right )-F\left ( 2 \right ) \right )
0%
\displaystyle \frac{1}{2}\left ( F\left ( 3 \right )-F\left ( 1 \right ) \right )
0%
\displaystyle 2\left ( F\left ( 6 \right )-F\left ( 2 \right ) \right )
Explanation
\displaystyle F\left ( x \right )=\int \frac{\sin x}{x}dx
Now
\displaystyle I = \int_{1}^{3}\frac{\sin 2x}{x}dx\left [ put2x=t \right ]=\int_{2}^{6}\frac{2}{2}\frac{\sin }{t}dt=\left [ F\left ( x \right ) \right ]_{2}^{6}=F(6)-F(2)
The value of
\displaystyle \int_{0}^{1}\displaystyle \frac{dx}{e^{x}+e^{-x}}
is
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0%
\tan ^{-1}e
0%
\tan ^{-1}\left ( e \right )-\pi /4
0%
\tan ^{-1}\left ( e \right )-\tan ^{-1}\left ( 1/e \right )
0%
\tan ^{-1}\left ( 1/e \right )+\pi /4
Explanation
Let
\displaystyle I=\int _{ 0 }^{ 1 } \frac { dx }{ e^{ x }+e^{ -x } } =\int _{ 0 }^{ 1 } \frac { { e }^{ x }dx }{ e^{ 2x }+1 }
Substitute
{ e }^{ x }=t\Rightarrow { e }^{ x }dx=dt
\displaystyle I=\int _{ 1 }^{ e }{ \frac { t }{ { t }^{ 2 }+1 } dt } ={ \left[ \tan ^{ -1 }{ t } \right] }_{ 1 }^{ e }=\tan ^{ -1 }{ e } -\frac { \pi }{ 4 }
\displaystyle \int_{1/2}^{2}\frac{1}{x}\sin \left ( x-\frac{1}{x} \right )dx
has the value equal to
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0%
0
0%
\displaystyle \frac{3}{4}
0%
\displaystyle \frac{5}{4}
0%
2
Explanation
\displaystyle I=\int _{ \frac { 1 }{ 2 } }^{ 2 }{ \frac { 1 }{ x } } \sin { \left( x-\frac { 1 }{ x } \right) dx }
Put
\displaystyle x=\frac { 1 }{ t } \Rightarrow dx=\frac { 1 }{ { t }^{ 2 } } dt
\displaystyle I=\int _{ 2 }^{ \frac { 1 }{ 2 } }{ t\sin { \left( \frac { 1 }{ t } -t \right) \left( \frac { -1 }{ { t }^{ 2 } } \right) dt } } =\int _{ 2 }^{ \frac { 1 }{ 2 } }{ \frac { 1 }{ t } \sin { \left( t-\frac { 1 }{ t } \right) dt } }
\displaystyle =-\int _{ \frac { 1 }{ 2 } }^{ 2 }{ \frac { 1 }{ t } } \sin { \left( t-\frac { 1 }{ t } \right) dt } =-I.
\Rightarrow 2I=0\quad \Rightarrow I=0
If
\displaystyle f\left ( x \right )=\int_{1}^{x}\frac{\ln t}{1+t}dt
where
x > 0
, then the value(s) of
x
satisfying the equation,
f(x) +f(1/x)=2
is
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0%
2
0%
e
0%
\displaystyle e^{-2}
0%
\displaystyle e^{2}
Explanation
\displaystyle f\left( x \right)=\int _{ 1 }^{ x }{ \frac { \ln { t } }{ 1+t } dt }
\displaystyle \Rightarrow f\left( \frac { 1 }{ x } \right) =\int _{ 1 }^{ 1/x }{ \cfrac { \ln { t } }{ 1+t } dt }
Substituting
t=\dfrac { 1 }{ u } \Rightarrow dt=\left( -\dfrac { 1 }{ { u }^{ 2 } } \right) du
Therefore
\displaystyle f\left( \dfrac { 1 }{ x } \right) =\int _{ 1 }^{ x }{ \dfrac { \ln { \left( \dfrac { 1 }{ u } \right) \left( -1 \right) } }{ \left( 1+\dfrac { 1 }{ u } \right) { u }^{ 2 } } du }
\displaystyle=\int _{ 1 }^{ x }{ \frac { \ln { u } }{ u\left( u+1 \right) } du } =\int _{ 1 }^{ x }{ \frac { \ln { t } }{ t\left( 1+t \right) } dt }
Now,
\displaystyle f\left( x \right) +f\left( \frac { 1 }{ x } \right) =\int _{ 1 }^{ x }{ \cfrac { \ln { t } }{ \left( 1+t \right) } dt } +\int _{ 1 }^{ x }{ \frac { \ln { t } }{ t\left( 1+t \right) } dt }
\displaystyle=\int _{ 1 }^{ x }{ \frac { \left( 1+t \right) \ln { t } }{ t\left( 1+t \right) } dt } =\int _{ 1 }^{ x }{ \frac { \ln { t } }{ t } dt } =\frac { 1 }{ 2 } { \left( \ln { x } \right) }^{ 2 }
Hence
\displaystyle f\left( x \right) +f\left( \frac { 1 }{ x } \right) =2\Rightarrow x={ e }^{ \pm 2 }
Solve
\displaystyle \int_{2}^{-13}\frac{dx}{\sqrt[5]{\left ( 3-x \right )^{4}}}
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0%
\displaystyle -5\left ( \sqrt[5]{16}-1 \right )
0%
\displaystyle 5\left ( \sqrt[5]{16}-1 \right )
0%
\displaystyle -5\left ( \sqrt[5]{16}+1 \right )
0%
None of these
Explanation
Let
\displaystyle I=\int _{ 2 }^{ -13 }{ \frac { dx }{ 5\sqrt { { \left( 3-x \right) }^{ 4 } } } }
Put
3-x=t\Rightarrow -dx=dt
\displaystyle I=\int _{ 1 }^{ 16 }{ \frac { dt }{ { t }^{ \frac { 4 }{ 5 } } } } ={ \left[ \frac { { t }^{ \frac { 1 }{ 5 } } }{ \frac { 1 }{ 5 } } \right] }_{ 1 }^{ 16 }
=-5\left( \sqrt [ 5 ]{ 16 } -1 \right)
Choose a function
f(x)
such that it is integrable over every interval on the real line
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0%
f(x) = [x]
0%
f(x)=x|x|
0%
f(x)=[sinx]
0%
f(x)=\dfrac{|x-1|}{x-1}
\displaystyle \int_{0}^{\infty }\frac{x}{\left ( 1+x \right )\left ( 1+x^{2} \right )}dx
Report Question
0%
\displaystyle \frac{\pi }{4}
0%
\displaystyle \frac{\pi }{2}
0%
is sme as
\displaystyle \int_{0}^{\infty }\frac{dx}{\left ( 1+x \right )\left ( 1+x^{2} \right )}
0%
cannot be evaluated
Explanation
Let
I=\int _{ 0 }^{ \infty }{ \cfrac { xdx }{ \left( 1+x \right) \left( 1+{ x }^{ 2 } \right) } }
Using partial fraction
=\int _{ 0 }^{ \infty }{ \left( \cfrac { x+1 }{ 2\left( 1+{ x }^{ 2 } \right) } -\cfrac { 1 }{ 2\left( 1+x \right) } \right) dx } \\ ={ \left( \lim _{ b\rightarrow \infty }{ \cfrac { 1 }{ 2 } \log { \left( 1+{ x }^{ 2 } \right) } -\cfrac { 1 }{ 2 } \log { \left( 1+x \right) } +\cfrac { 1 }{ 2 } { tan }^{ -1 }x } \right) }_{ 0 }^{ b }\\ =\lim _{ b\rightarrow \infty }{ \left( \cfrac { 1 }{ 2 } \log { \left( 1+b \right) } -\cfrac { 1 }{ 2 } \log { \left( 1+b \right) } +\cfrac { 1 }{ 2 } { tan }^{ -1 }b \right) } \\ =\cfrac { \pi }{ 4 }
State true or false:
The average value of the function
f(x) = sin^2xcos^3x
on the interval
[ -\pi ,\pi ]
is 0.
Report Question
0%
True
0%
False
Explanation
\int _{ -\pi }^{ \pi }{ f\left( x \right) dx } =\int _{ -\pi }^{ \pi }{ \sin ^{ 2 }{ x } \cos ^{ 3 }{ x } dx } \\ =\int _{ -\pi }^{ \pi }{ \sin ^{ 2 }{ x } \left( 1-\sin ^{ 2 }{ x } \right) \cos { x } dx } \\ =\int _{ -\pi }^{ \pi }{ \sin ^{ 2 }{ x } \cos { x } dx } -\int _{ -\pi }^{ \pi }{ \sin ^{ 4 }{ x } \cos { x } dx }
\displaystyle ={ \left[ \frac { \sin ^{ 3 }{ x } }{ 3 } \right] }_{ -\pi }^{ \pi }-{ \left[ \frac { \sin ^{ 5 }{ x } }{ 3 } \right] }_{ -\pi }^{ \pi }=0
I_{1}
is equal to
Report Question
0%
\displaystyle \frac {2}{3} \int_{0}^{\pi/2}(sin^{2}\theta)(cos\theta)^{-1/3}d\theta
0%
\displaystyle \frac {3}{2} \int_{0}^{\pi/2}(sin^{2}\theta)(cos\theta)^{-1/3}d\theta
0%
\displaystyle \frac {2}{3} \int_{0}^{\pi/2}(sin\theta)^{2/3}(cos\theta)^{-1/3}d\theta
0%
\displaystyle \frac {3}{2} \int_{0}^{\pi/2}(sin\theta)^{2/3}(cos\theta)^{-1/3}d\theta
Explanation
\displaystyle I_{1}=\int_{0}^{1}(1-x^{2})^{1/3}dx=(1-x^{2})^{3}_{0}+\frac {2}{3} \int_{0}^{1}\frac {x^{2}}{(1-x^{2})^{2/3}}dx
x=sin\theta, dx=cos\theta d\theta
\displaystyle I_{1}=\frac {2}{3} \int_{0}^{\pi/2}\frac {sin^{2}\theta cos\theta}{(cos\theta)^{4/3}}d\theta
\displaystyle I_{1}=\frac {2}{3} \int_{0}^{\pi/2}sin^{2}\theta (cos\theta)^{-1/3}d\theta
Ans:
A
Evaluate
\displaystyle \int_{0}^{\pi /2} \frac{dx}{2+\sin 2x}
Report Question
0%
\displaystyle \frac{2\pi }{{3}}
0%
\displaystyle \frac{\pi }{{3}}
0%
\displaystyle \frac{2\pi }{{5}}
0%
None of these
Explanation
\displaystyle \int _{ 0 }^{ 2\pi }{ \frac { dx }{ 2+\sin { 2x } } } =\int _{ 0 }^{ 2\pi }{ \frac { dx }{ 2+\frac { 2\tan { x } }{ 1+\tan ^{ 2 }{ x } } } } dx
\displaystyle =\frac { 1 }{ 2 } \int _{ 0 }^{ 2x }{ \frac { \sec ^{ 2 }{ x } }{ \tan ^{ 2 }{ x } +\tan { x+1 } } dx }
Put
\tan { x } =t\Rightarrow \sec ^{ 2 }{ x } dx=dt
\displaystyle I=\frac { 1 }{ 2 } \int { \frac { 1 }{ { { t }^{ 2 }+t+1 } } } dt.=\frac { 1 }{ 2 } \int { \frac { 1 }{ { \left( t+\frac { 1 }{ 2 } \right) }^{ 2 }-\frac { 3 }{ 2 } } } =\frac { 2\pi }{ 3 }
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0%
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
0%
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
0%
Assertion is correct but Reason is incorrect
0%
Both Assertion and Reason are incorrect
Explanation
Reason is true
Assertion
Put
\displaystyle x=\frac{1}{t}\Rightarrow dx=-\frac{1}{t^{2}}dt
\displaystyle l=-\int_{3}^{1/3}tcosec ^{99}\left ( \frac{1}{t}-t \right )\frac{1}{t^{2}}dt
\displaystyle l=-\int_{1/3}^{3}\frac{1}{t}cosec^{99}\left ( t-\frac{1}{t} \right )dt
\displaystyle l=-1
\Rightarrow 2l=0
\Rightarrow l=0
\displaystyle \int_{\tfrac{1}{\sqrt{3}}}^{0}\dfrac{dx}{\left ( 2x^{2}+1 \right )\sqrt{x^{2}+1}}
Report Question
0%
\displaystyle - \tan^{-1} \dfrac{1}{2}
0%
\displaystyle \tan^{-1} 1
0%
\displaystyle - \tan^{-1} \dfrac{1}{3}
0%
\displaystyle \tan^{-1} \dfrac{1}{\sqrt 2}
Explanation
I=\displaystyle \int_{\frac{1}{\sqrt{3}}}^{0}\frac{dx}{\left ( 2x^{2}+1 \right )\sqrt{x^{2}+1}}
Substitute
x=\displaystyle \frac{1}{z}
\Rightarrow\displaystyle dx=-\frac{1}{z^2}dz
I=\displaystyle \int _{ \sqrt { 3 } }^{ \infty } \dfrac { -\dfrac { 1 }{ { z }^{ 2 } } dz }{ \left( \dfrac { 2 }{ { z }^{ 2 } } +1 \right) \sqrt { \dfrac { 1 }{ { z }^{ 2 } } +1 } }
=\displaystyle \int _{ \sqrt { 3 } }^{ \infty } \dfrac { -zdz }{ \left( 2+{ z }^{ 2 } \right) \sqrt { 1+{ z }^{ 2 } } }
Substitute
\sqrt { 1+{ z }^{ 2 } } =t
\Rightarrow z^2+1=t^2
\Rightarrow zdz=tdt
I=\displaystyle \int _{ 2 }^{ \infty } \frac { -tdt }{ \left( 1+{ t }^{ 2 } \right) t }
I=[\cot^{-1}t]_{2}^{\infty}
\Rightarrow I=-\cot^{-1}2
\Rightarrow I=-\tan^{-1}{\dfrac{1}{2}}
The value of definite integral
\displaystyle \int_{\infty }^{0}\frac{Ze^{-z}}{\sqrt{1-e^{-2z}}}dz
Report Question
0%
\displaystyle -\frac{\pi }{2}ln2
0%
\displaystyle \frac{\pi }{2}ln2
0%
\displaystyle -\pi ln2
0%
\displaystyle \pi ln\frac{1}{\sqrt{2}}
Explanation
\displaystyle l=\int_{\infty }^{0}\frac{ze^{-z}}{\sqrt{1-e^{-2z}}}dz
put
\displaystyle e^{-z}=\sin \theta
\displaystyle l=-\int_{0}^{\pi /2}\frac{ln\left ( \sin \theta \right )\left ( -\cos \theta \right )d\theta }{\sqrt{1-\sin ^{2}\theta }}=\int_{0}^{\pi /2}ln\sin \theta d\theta
\displaystyle \frac{-\pi }{2}ln2
If
\displaystyle \int _{ 0 }^{ 1 }{ { e }^{ { x }^{ 2 } }\left( x-\alpha \right) } dx=0
, then
Report Question
0%
1<\alpha <2
0%
\alpha <2
0%
0<\alpha<1
0%
\alpha=0
Explanation
\displaystyle\because \int _{ 0 }^{ 1 }{ { e }^{ { x }^{ 2 } }\left( x-\alpha \right) } dx=0,
\therefore { e }^{ { x }^{ 2 } }\left( x-\alpha \right)
must be
+ve
and
-ve
both for
x\in (0,1)
i.e.
{ e }^{ x }\left( x-\alpha \right) =0
for one
x\in (0,1)
\therefore \alpha \in(0,1)
\displaystyle \int _{ 0 }^{ a }{ \frac { dx }{ a+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } }
is equal to
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0%
\displaystyle \frac { \pi }{ 2 } +1
0%
\displaystyle \frac { \pi }{ 2 } -1
0%
\displaystyle 1-\frac { \pi }{ 2 }
0%
none of these
Explanation
Let
\displaystyle I=\int _{ 0 }^{ a }{ \frac { dx }{ a+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } } =\int _{ 0 }^{ \pi /2 }{ \frac { a\cos { \theta } d\theta }{ a+a\cos { \theta } } }
By putting
x=a\sin { \theta } \Rightarrow dx=a\cos { \theta } d\theta
\displaystyle I=\int _{ 0 }^{ \pi /2 }{ \frac { \cos { \theta } }{ 1+\cos { \theta } } d\theta } =\int _{ 0 }^{ \pi /2 }{ \left( 1-\frac { 1 }{ 1+\cos { \theta } } \right) d\theta }
\displaystyle =\int _{ 0 }^{ \pi /2 }{ d\theta } -\frac { 1 }{ 2 } \int _{ 0 }^{ \pi /2 }{ \sec ^{ 2 }{ \frac { \theta }{ 2 } } d\theta }
\displaystyle ={ \left[ \theta -\tan { \frac { \theta }{ 2 } } \right] }_{ 0 }^{ \pi /2 }=\frac { \pi }{ 2 } -\tan { \frac { \pi }{ 4 } } =\left( \frac { \pi }{ 2 } -1 \right)
What is
\displaystyle\int _{ 1 }^{ 2 }{ \ln { x } dx }
equal to?
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0%
\ln { 2 }
0%
1
0%
\ln { \left( \dfrac { 4 }{ e } \right) }
0%
\ln { \left( \dfrac { e }{ 4 } \right) }
Explanation
Let,
y=\ln x
then,
x={ e^{ y } }
Now at,
x=1,y=0
x=2,y=\ln 2
\Rightarrow dx={ e }^{ y }.dy
Putting the values, we get
\displaystyle \int_{1}^{2} \ln x dx=\int _{ 0 }^{ \ln2 }{ { e }^{ y }.y.dy }
=\displaystyle { \left[ y.{ e }^{ y } \right] }_{ 0 }^{\ln2 }-\int _{ 0 }^{\ln2 }{ { e }^{ y }dy }
=2\ln 2-1
=2\ln 2-\ln(e)
=\ln\left(\dfrac { 4 }{ e }\right)
Hence, C is correct.
\displaystyle\int_0^\pi \frac{1}{1\, +\,\sin x}\ dx
is equal to
Report Question
0%
1
0%
2
0%
3
0%
4
Explanation
Let
I = \displaystyle\int_0^\pi \frac{1}{1\, +\,\sin x}\, dx\,
=\, \displaystyle \int_0^\pi \frac{1}{1\, +\,\displaystyle \frac{2 \tan \frac {x}{2}}{1\, +\, \tan^2 \frac {x}{2}}}\, dx
=\, \displaystyle\int_0^\pi \frac{\sec^2 \dfrac {x}{2}}{\left(1\, +\,\tan \dfrac {x}{2}\right)^2}\, dx
Put
1+\tan
\dfrac{x}{2}\,= t\,\Rightarrow\,\dfrac{1}{2}\,\sec^2\,\dfrac{x}{2}\,=\,dt
\therefore\,I\,=\,\displaystyle\int_1^\infty\dfrac{2dt}{1\,+\,t^2}\,=-\dfrac2t\bigg]_1^\infty = 2
The value of
\displaystyle\int _{ 0 }^{ { x }/{ 4 } }{ \dfrac { \sec { x } }{ { \left( \sec { x } +\tan { x } \right) }^{ 2 } } dx }
is
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0%
1+\sqrt{2}
0%
-11+\sqrt{2}
0%
-\sqrt{2}
0%
None of these
Explanation
I=\displaystyle \int _{ 0 }^{ \frac { \pi }{ 4 } }{ \cfrac { \sec { x } }{ { \left( \sec { x } +\tan { x } \right) }^{ 2 } } } dx
=\displaystyle \int _{ 0 }^{ \frac { \pi }{ 4 } }{ \cfrac { { 1 }/{ \cos { x } } }{ { \left[ \cfrac { 1 }{ \cos { x } } +\cfrac { \sin { x } }{ \cos { x } } \right] }^{ 2 } } } dx
=\displaystyle \int _{ 0 }^{ \frac { \pi }{ 4 } }{ \cfrac { { \cos ^{ 2 }{ x } }/{ \cos { x } } }{ { \left( 1+\sin { x } \right) }^{ 2 } } } dx
I=\displaystyle \int _{ 0 }^{ \frac { \pi }{ 4 } }{ { \left( 1+\sin { x } \right) }^{ -2 } } \cos { x } dx
Take
1+\sin x=t \implies \cos x dx=dt
When
x=0, t=1
and
x=\dfrac{\pi}{4}, t=1+\dfrac{1}{\sqrt{2}}
\therefore \quad I=\int_{1}^{1+\frac{1}{\sqrt{2}}} t^{-2}\ dt
\therefore \quad I=\left|\dfrac{t^{-1}}{-1}\right|_{1}^{1+\frac{1}{\sqrt{2}}}
=-\left[ \cfrac { 1 }{ 1+\frac { 1 }{ \sqrt { 2 } } } -1 \right]
=-\cfrac { \sqrt { 2 } }{ \sqrt { 2 } +1 } +1
=\cfrac { -\sqrt { 2 } +\sqrt { 2 } +1 }{ \sqrt { 2 } +1 } =\cfrac { 1 }{ \sqrt { 2 } +1 }
=\cfrac { 1 }{ \sqrt { 2 } +1 } \ast \cfrac { \sqrt { 2 } -1 }{ \sqrt { 2 } -1 } =\cfrac { \sqrt { 2 } -1 }{ 2-1 }
=\sqrt { 2 } -1
\therefore
None of these is correct option
If
\quad f(x)=\begin{cases} 2{ x }^{ 2 }+1,x\le 1 \\ 4{ x }^{ 2 }-1,x>1 \end{cases}
, then
\int _{ 0 }^{ 2 }{ f(x)dx }
is
Report Question
0%
10
0%
50/3
0%
1/3
0%
47/2
Explanation
\quad f(x)=\begin{cases} 2{ x }^{ 2 }+1,x\le 1 \\ 4{ x }^{ 2 }-1,x>1 \end{cases}
from 0 to 1 we have
2x^2+1
and from 1 to 2 we have
4x^2-1
=\int _{ 0 }^{ 1 }{ \left( 2{ x }^{ 2 }+1 \right) dx+\int _{ 1 }^{ 2 }{ \left( 4{ x }^{ 2 }-1 \right) dx } }
={ \left[ \frac { 2{ x }^{ 3 } }{ 3 } +x \right] }_{ 0 }^{ 1 }+{ \left[ \frac { 4{ x }^{ 3 } }{ 3 } -x \right] }_{ 1 }^{ 2 }
=\frac { 2 }{ 3 } +1-0-0+\frac { 4{ (2) }^{ 3 } }{ 3 } -2-\left( \frac { 4 }{ 3 } -1 \right)
=\frac { 2 }{ 3 } +1+\frac { 32 }{ 3 } -2-\frac { 4 }{ 3 } +1
=\frac { 2+32-4 }{ 3 }
=\dfrac{30}{3}
=10.
What is
\displaystyle \int_0^1 {\frac{\tan^{-1}x}{1 + x^2} dx}
equal to ?
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0%
\dfrac{\pi}{4}
0%
\dfrac{\pi}{8}
0%
\dfrac{\pi^2}{8}
0%
\dfrac{\pi^2}{32}
Explanation
Let
I=\displaystyle \int _{ 0 }^{ 1 } \dfrac { {\tan^{ -1 } }x }{ { 1+x^{ 2 } } } dx
Let,
y={ {\tan^{ -1 } }x }
\Rightarrow dy=\dfrac { 1 }{ { 1+x^{ 2 } } } dx
Now for
x=1, y=\dfrac { \pi }{ 4 }
and for
x=0, y=0
Putting the values in the integral we have,
I=\displaystyle \int _{ 0 }^{ \frac { \pi }{ 4 } }{ y.dy }
= { \left[\dfrac { y^{ 2 } }{ 2 } \right] }_{ 0 }^{ \frac { \pi }{ 4 } }
=\dfrac { \pi ^{ 2 } }{ 32 }
Hence, D is correct.
If
\dfrac {dy}{dt} = ky
and
k\neq 0
, which of the following could be the equation of
y
?
Report Question
0%
y = kx - 7
0%
y = 95e^{kt}
0%
y = 5 + ln\ k
0%
y = (x - k)^{2}
0%
y = \sqrt [k]{x}
Explanation
Given,
\dfrac{dy}{dt} = ky
and
k \ne 0
We have
\dfrac{dy}{y} = kdt
Apply integral on both sides, so we get
\displaystyle \int \dfrac{dy}{y} = \int kdt
\Rightarrow \ln(y) = kt+c
(where
c
is constant)
\Rightarrow y = {e}^{c}{e}^{kt} = a{e}^{kt}
(
a = {e}^{c}
)
The possible solution is
y = 95{e}^{kt}
Solve
\int_{0}^{\dfrac {\pi}{2}}\sqrt {\sin \phi}\cos^{5}\phi d\phi
.
Report Question
0%
\dfrac{64}{231}
0%
\dfrac{24}{231}
0%
\dfrac{54}{231}
0%
None of these
Explanation
Let
\begin{matrix} I=\int _{ 0 }^{ \dfrac { \pi }{ 2 } }{ \sqrt { \sin \phi } { { \cos }^{ 5 } }\phi d\phi } \\ =\int _{ 0 }^{ \dfrac { \pi }{ 2 } }{ \sqrt { \sin \phi } { { \cos }^{ 4 } }\phi \cos \phi d\phi } \\ =\int _{ 0 }^{ \dfrac { \pi }{ 2 } }{ \sqrt { \sin \phi } { { \left( { 1-{ { \sin }^{ 2 } }\phi } \right) }^{ 2 } }\cos \phi d\phi } \\ put\, \, \, t=\sin \phi \\ \dfrac { { dt } }{ { d\phi } } =\cos \phi \\ dt=\cos \phi d\phi \\ When\, \, \phi =0,\, \, t=0\, \, when\, \, \phi =\dfrac { \pi }{ 2 } ,\, t=1 \\ \therefore I=\int _{ 0 }^{ 1 }{ \sqrt { t } { { \left( { 1-{ t^{ 2 } } } \right) }^{ 2 } }dt } \\ =\int _{ 0 }^{ 1 }{ \sqrt { t } \left( { 1-2{ t^{ 2 } }+{ t^{ 4 } } } \right) dt } \\ =\int _{ 0 }^{ 1 }{ { t^{ \dfrac { 1 }{ 2 } } }+{ t^{ \dfrac { 9 }{ 2 } } }-2{ t^{ \dfrac { 5 }{ 2 } } }dt } \\ =\left[ { \dfrac { 2 }{ 3 } { t^{ \dfrac { 3 }{ 2 } } }+\dfrac { 2 }{ { 11 } } { t^{ \dfrac { { 11 } }{ 2 } } }-\dfrac { 4 }{ 7 } { t^{ \dfrac { 7 }{ 2 } } } } \right] _{ 0 }^{ 1 } \\ =\dfrac { 2 }{ 3 } +\dfrac { 2 }{ { 11 } } -\dfrac { 4 }{ 7 } \\ =\dfrac { { 154+42-132 } }{ { 3\times 11\times 7 } } =\dfrac { { 64 } }{ { 231 } } \\ \end{matrix}
\displaystyle\int _{ 0 }^{ { \sqrt { \pi } }/{ 2 } }{ 2{ x }^{ 3 }\sin { \left( { x }^{ 2 } \right) } dx }
is equal to
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0%
\dfrac { 1 }{ \sqrt { 2 } } \left( 1+\dfrac { \pi }{ 4 } \right)
0%
\dfrac { 1 }{ \sqrt { 2 } } \left( 1-\dfrac { \pi }{ 4 } \right)
0%
\dfrac { 1 }{ \sqrt { 2 } } \left( \dfrac { \pi }{ 2 } -1 \right)
0%
\dfrac { 1 }{ \sqrt { 2 } } \left( 1-\dfrac { \pi }{ 2 } \right)
0%
\dfrac { 1 }{ \sqrt { 2 } } \left( \dfrac { \pi }{ 4 } -1 \right)
Explanation
I=\displaystyle\int _{ 0 }^{ { \sqrt { \pi } }/{ 2 } }{ 2{ x }^{ 3 }\sin { \left( { x }^{ 2 } \right) } dx }
=\displaystyle\int _{ 0 }^{ { \sqrt { \pi } }/{ 2 } }{ 2{ x }^{ 2 }\cdot x\sin { \left( { x }^{ 2 } \right) } dx }
Put
{ x }^{ 2 }=t
\Rightarrow 2xdx=dt
Also, when
x=0
, then
t=0
and when
x=\dfrac { \sqrt { \pi } }{ 2 }
, then
t=\dfrac { \pi }{ 4 }
\Rightarrow I=\displaystyle\int _{ 0 }^{ { \pi }/{ 4 } }{ \underset { I }{ t } \underset { II }{ \sin { t } } dt }
={ \left[ t\left( -\cos { t } \right) \right] }_{ 0 }^{ { \pi }/{ 4 } }-\displaystyle\int _{ 0 }^{ { \pi }/{ 4 } }{ -\cos { t } \left( 1 \right) dt }
={ \left[ -t\cos { t } \right] }_{ 0 }^{ { \pi }/{ 4 } }+\displaystyle\int _{ 0 }^{ { \pi }/{ 4 } }{ \cos { t } dt }
={ \left[ -t\cos { t } +\sin { t } \right] }_{ 0 }^{ { \pi }/{ 4 } }
=\left[ -\dfrac { \pi }{ 4 } \cdot \dfrac { 1 }{ \sqrt { 2 } } +\dfrac { 1 }{ \sqrt { 2 } } \right]
=\dfrac { 1 }{ \sqrt { 2 } } \left( 1-\dfrac { \pi }{ 4 } \right)
\displaystyle\int { \cfrac { 1 }{ 7 } \sin { \left( \cfrac { x }{ 7 } +10 \right) } dx }
is equal to
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0%
\cfrac { 1 }{ 7 } \cos { \left( \cfrac { x }{ 7 } +10 \right) } +C
0%
-\cfrac { 1 }{ 7 } \cos { \left( \cfrac { x }{ 7 } +10 \right) } +C
0%
-\cos { \left( \cfrac { x }{ 7 } +10 \right) } +C
0%
-7\cos { \left( \cfrac { x }{ 7 } +10 \right) } +C
0%
\cos { \left( x+70 \right) } +C
Explanation
Let
I=\int { \cfrac { 1 }{ 7 } \sin { \left( \cfrac { x }{ 7 } +10 \right) } dx }
=\cfrac { 1 }{ 7 } \int { \sin { \left( \cfrac { x }{ 7 } +10 \right) } } =\cfrac { 1 }{ 7 } \cfrac { -\cos { \left( \cfrac { x }{ 7 } +10 \right) } }{ \cfrac { 1 }{ 7 } }
=-\cos { \left( \cfrac { x }{ 7 } +10 \right) } +C
If
\int _0^1 xdx = \dfrac {\pi}{4} - \dfrac {1}{2} ln 2
then the value of definite integral
\int _0^1 \tan^{-1} (1-x+x^2) dx
equals :
Report Question
0%
ln2
0%
\dfrac {\pi}{4} + ln 2
0%
\dfrac {\pi}{4} - ln2
0%
2 ln 2
The value of
\int_{0}^{1} \dfrac {8\log (1 + x)}{1 + x^{2}} dx
is
Report Question
0%
\dfrac {\pi}{2}\log 2
0%
\pi\log 2
0%
2\pi\log 2
0%
None of these
Explanation
Let
\begin{matrix} I=\int _{ 0 }^{ 1 }{ \frac { { 8\log \left( { 1+x } \right) } }{ { 1+{ x^{ 2 } } } } } dx \\ \, \, \, \, =8\int _{ 0 }^{ 1 }{ \frac { { \log \left( { 1+x } \right) } }{ { 1+{ x^{ 2 } } } } } dx \\ put\, \, \, x=\tan \theta \\ \, \, \, \, \, \Rightarrow dx={ \sin ^{ 2 } }\theta d\theta \\ x,\, \frac { \pi }{ 4 } ,\, \, \, \, x=0 \\ \quad \quad \quad \quad I=8\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \frac { { \log \left( { 1+x } \right) } }{ { { { \sin }^{ 2 } }\theta } } } { \sin ^{ 2 } }\theta d\theta \\ \, \, \, \, \, \quad \quad \quad \quad =8\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \log \left( { 1+\tan x } \right) } dx \\ \quad \quad =8\times \frac { \pi }{ 8 } \log 2 \\ \, \, \, =\pi \log 2 \\ \end{matrix}
Hence option (A) is true.
If
\displaystyle\int { \sqrt { 1+\sin { x } } \cdot f\left( x \right) dx } =\dfrac { 2 }{ 3 } { \left( 1+\sin { x } \right) }^{ { 3 }/{ 2 } }+C
, then
f\left( x \right)
is equal to
Report Question
0%
\cos { x }
0%
\sin { x }
0%
\tan { x }
0%
1
Explanation
\int { \sqrt { 1+\sin { x } } } .f\left( x \right) dx=\dfrac { 2 }{ 3 } { (1+\sin { x } ) }^{ { 3 }/{ 2 } }+c
On differentiating both sides, we get
\sqrt { 1+\sin { x } } .f\left( x \right) =\dfrac { 2 }{ 3 } .\dfrac { 3 }{ 2 } { (1+\sin { x } ) }^{ { 1 }/{ 2 } }.\cos { x } +0\\ \Rightarrow f\left( x \right) =\cos { x }
\displaystyle \int _0^{\pi /2} f(\sin 2x)\sin x\, dx = K\int_0^{\pi/2} f(\cos 2x) \cos x\,dx
where
k
equals to
Report Question
0%
2
0%
4
0%
\sqrt{2}
0%
2\sqrt{2}
Explanation
\displaystyle \int_0^{\dfrac{\pi}{2}} \sin 2x \sin x dx = K \int_0^{\dfrac{\pi}{2}} \cos 2x \cos x dx
\displaystyle \int_0^{\dfrac{\pi}{2}} 2 \sin x \cos x \sin x dx = K \int_0^{\dfrac{\pi}{2}} (1 - 2 \sin^2 x) \cos x dx
\displaystyle \int_0^{\dfrac{\pi}{2}} 2 \sin^2 x \cos x dx = K \left[\int_0^{\dfrac{\pi}{2}} \cos x dx - \int_0^{\dfrac{\pi}{2}} 2 \sin^2 x \cos x dx \right]
\left\{\dfrac{2 \sin^3 x}{3} \right\}_0^{\dfrac{\pi}{2}} = K \left[\{\sin x\}_0^{\dfrac{\pi}{2}} - \left\{\dfrac{2 \sin^3 x}{3} \right\}_0^{\dfrac{\pi}{2}}\right]
\dfrac{2}{3} = K \left[1 - \dfrac{2}{3} \right]
K = 2
\int_{0}^{1}{\frac{dx}{x\sqrt{x}}}
Report Question
0%
2
0%
-2
0%
1
0%
3
Explanation
\int _{ 0 }^{ 1 }{ \cfrac { 1 }{ x\sqrt { x } } dx } \\ \int _{ 0 }^{ 1 }{ { x }^{ \cfrac { -3 }{ 2 } }dx } ={ [\cfrac { { x }^{ \cfrac { -3 }{ 2 } +1 } }{ \cfrac { -3 }{ 2 } +1 } ] }_{ 0 }^{ 1 }\\ ={ [\cfrac { { x }^{ \cfrac { -1 }{ 2 } } }{ \cfrac { -1 }{ 2 } } ] }_{ 0 }^{ 1 }=\cfrac { 1-0 }{ \cfrac { -1 }{ 2 } } =-2
What is
\displaystyle \int_{0}^{2\pi}\sqrt {1 + \sin \dfrac {x}{2}}dx
equal to?
Report Question
0%
8
0%
4
0%
2
0%
0
Explanation
1+\sin\frac { x }{ 2 } ={ \sin }^{ 2 }\frac { x }{ 4 } +{ \cos }^{ 2 }\frac { x }{ 4 } +2\sin\frac { x }{ 4 } \cos\frac { x }{ 4 } \\ { (\sin\frac { x }{ 4 } +\cos\frac { x }{ 4 } ) }^{ 2 }\\ \sqrt { 1+\sin\frac { x }{ 2 } } =\sqrt { { (\sin\frac { x }{ 4 } +\cos\frac { x }{ 4 } ) }^{ 2 } } =\sin\frac { x }{ 4 } +\cos\frac { x }{ 4 }
\displaystyle \int _{ 0 }^{ 2\pi }{ \left(\sin\frac { x }{ 4 } +\cos\frac { x }{ 4 }\right )dx } =4\left(-\cos\frac { x }{ 4 } +\sin\frac { x }{ 4 }\right )+c\\ After\quad putting\quad the\quad limit\quad we\quad get\\\Rightarrow 4(1-(-1))=8\\
So correct answer will be option A
Let
I=\displaystyle \int _{ \pi /4 }^{ \pi /3 }{ \cfrac { \sin { x } }{ x } } dx
. Then?
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0%
\cfrac { 1 }{ 2 } \le I\le 1\quad
0%
4\le I\le 2\sqrt { 30 }
0%
\cfrac { \sqrt { 3 } }{ 8 } \le I\le \cfrac { \sqrt { 2 } }{ 6 }
0%
1\le I\le \cfrac { 2\sqrt { 3 } }{ \sqrt { 2 } }
Explanation
I=\displaystyle \int^{\tfrac{\pi}{3}}_{\tfrac{\pi}{4}}\dfrac{sinx}{x}dx
\dfrac{sinx}{x}
is a decreasing function in given interval
difference of limits
=\dfrac{\pi}{3}-\dfrac{\pi}{4}=\dfrac{\pi}{12}
so,
\dfrac{\pi}{12}\cdot\dfrac{sin\dfrac{\pi}{3}}{\dfrac{\pi}{3}} \le I \le \dfrac{\pi}{12} \cdot\dfrac{sin\dfrac{\pi}{4}}{\dfrac{\pi}{4}}
\dfrac{\sqrt3}{8} \leq I \leq \dfrac{\sqrt2}{6}
\int { { \left( ex \right) }^{ x }\left( 2+\log { x } \right) } dx=....+c,x\in { R }^{ + }-\left\{ 1 \right\}
Report Question
0%
{ x }^{ x }
0%
{ \left( ex \right) }^{ x }\quad
0%
{ e }^{ x }
0%
\left( 1+\log { x } \right) { \left( ex \right) }^{ x }
Explanation
\int (ex)^x(2+\log x)dx=\int (ex)^xdx+\int e^x.x^x(1+\log x)dx
Integrating by parts ,
\Rightarrow \int(ex)^x +e^x(\int x^x(1+\log x)) -\int(\dfrac{d}{dx}(e^x).(\int x^x(1+\log x)dx))dx
\Rightarrow \int (ex)^x+ e^xx^x-\int (ex)^x
\Rightarrow (ex)^x
If
I=\displaystyle\overset{1}{\underset{0}{\displaystyle\int}}x(1-x)^{1/2}dx
and
60I+k=25
then
k=
_________.
(k\in R)
.
Report Question
0%
9
0%
25
0%
60
0%
41
Explanation
We know,
\displaystyle \int_0^af(x)dx=\int_0^af(a-x)dx
\therefore I=\displaystyle \int_0^1x(1-x)^{1/2}dx=\int_0^1(1-x)x^{1/2}dx
\Rightarrow I=\displaystyle \int_0^1(x^{1/2}-x^{3/2})dx
\Rightarrow I=\left[\dfrac{2}{3}x^{3/2}-\dfrac{2}{5}x^{5/2} \right]_0^1
\Rightarrow I=\dfrac{2}{3}-\dfrac{2}{5}=\dfrac{4}{15}
60I+k=25
k=25-60\times \dfrac{4}{15}=9
\displaystyle \int_1^{32}\dfrac{dx}{x^{1/5}\sqrt{1+x^{4/5}}}
Report Question
0%
\dfrac{2}{5}(\sqrt{17}+\sqrt{2})
0%
\dfrac{2}{5}(\sqrt{17}-\sqrt{2})
0%
\dfrac{5}{2}(\sqrt{17}-\sqrt{2})
0%
\dfrac{5}{2}(\sqrt{17}+\sqrt{2})
\int _{ 0 }^{ \pi /2 }{ \sin { 2x } .\sin { x } } dx=.....
Report Question
0%
\cfrac{1}{3}
0%
\cfrac{2}{3}
0%
-\cfrac{2}{3}
0%
\cfrac{4}{3}
Explanation
I=\int _{ 0 }^{ \pi /2 }{ \sin { 2x } .\sin { x } } dx
=\int _{ 0 }^{ \pi /2 }{ 2\sin { x } \cos { x } .\sin { x } } dx=2\int _{ 0 }^{ \pi /2 }{ \sin ^{ 2 }{ x } \cfrac { d }{ dx } \sin { x } } dx
=\cfrac { 2 }{ 3 } { \left[ \sin ^{ 2 }{ x } \right] }_{ 0 }^{ \pi /2 }=\cfrac { 2 }{ 3 } (1-0)=\cfrac { 2 }{ 3 }
Evaluate
\displaystyle \int_{-2\pi}^{5\pi} \cot^{-1} (\tan x) dx
.
Report Question
0%
0
0%
-1
0%
1
0%
2
Explanation
I=\displaystyle \int_{2\pi} ^{5\pi}\cot ^{-1}(\tan x)dx
......
(1)
Applying property of limits of integral formula ;
\Rightarrow I=\displaystyle \int_{2\pi} ^{5\pi}\cot ^{-1}(\tan (7\pi - x)) dx
\Rightarrow I=\displaystyle \int_{2\pi} ^{5\pi}\cot ^{-1}(-\tan x)dx
\Rightarrow I=\displaystyle -\int_{2\pi} ^{5\pi}\cot ^{-1}(\tan x)dx
......
(2)
Adding both equations;
\Rightarrow 2I=0
\Rightarrow I=0
\Rightarrow \displaystyle \int_{2\pi} ^{5\pi}\cot ^{-1}(\tan x)dx=0
\int {\left( 3.{ x }^{ 2 }.\tan ^{ -1 }{ x } +\cfrac { { x }^{ 3 } }{ 1+{ x }^{ 2 } } \right) } dx=....+c
.
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0%
{ x }^{ 3 }\tan ^{ -1 }{ x }
0%
\cfrac { { x }^{ 3 } }{ 3 } \tan ^{ -1 }{ x }
0%
{ x }^{ 2 }\tan ^{ -1 }{ x }
0%
\cfrac { { x }^{ 2 } }{ 2 } \tan ^{ -1 }{ x }
Explanation
Given
\int (3x^2.\tan^{-1}x+\dfrac{x^3}{1+x^2})dx
=
\int (3x^2.\tan^{-1}x.dx)+\int(\dfrac{x^3}{1+x^2})dx
Integrating by parts
\int 3x^2tan^{-1}x.dx+x^3\int\dfrac{1}{1+x^2}dx-\int( \dfrac{d}{dx}(x^3)(\int \dfrac{1}{1+x^2}dx)dx)
\dfrac{d}{dx}x^3=3x^2;\int \dfrac{1}{1+x^2}=tan^{-1}x
\Rightarrow\int 3x^2tan^{-1}x.dx + x^3 \tan^{-1}x-\int 3x^2tan^{-1}x.dx
\Rightarrow x^3\tan^{-1}x+c
If
\displaystyle \int _0^{\pi/2} \sin x \cos x dx
is equal to:
Report Question
0%
\dfrac 1 2
0%
\dfrac 14
0%
2
0%
1
Explanation
\displaystyle \int _0^{\pi/2} \sin x \cos x dx
\sin x=t\implies \cos x dx=dt
x\to 0\to \dfrac \pi 2
t\to 0\to 1
\Rightarrow \displaystyle \int _0^{\pi/2} t dt
\Rightarrow\left.\dfrac {t^2}2\right|^1_0
\Rightarrow\dfrac 12-0=\dfrac 12
I= \int \frac{x+2}{(x+1)^2}dx;
then I is equal to
Report Question
0%
\log (x+1)+\dfrac{1}{x+1}+c
0%
\log (x+2)-\dfrac{1}{x+1}+c
0%
\log (1+x)-\dfrac{1}{x+1}+c
0%
\log (x+2)+\dfrac{1}{x+1}+c
Explanation
\int { \dfrac { x+2 }{ { \left( x+1 \right) }^{ 2 } } } dx
=\int { \dfrac { x+1+1 }{ { \left( x+1 \right) }^{ 2 } } } dx
=\int { \left( \dfrac { x+1 }{ { \left( x+1 \right) }^{ 2 } } +\dfrac { 1 }{ { \left( x+1 \right) }^{ 2 } } \right) } dx
=\int { \dfrac { 1 }{ x+1 } } dx+\int { \dfrac { 1 }{ { \left( x+1 \right) }^{ 2 } } } dx
\left[ \because \int { { x }^{ n }dx=\dfrac { { x }^{ n+1 } }{ n+1 } +c } \right]
=\log \left(x+1\right) +\dfrac { { \left( x+1 \right) }^{ -2+1 } }{ -2+1 } +c
\left[ \because \int { \dfrac { 1 }{ x } dx=\log { x+c } } \right]
=\log\left(x+1\right) +\left(-1\right) \dfrac{1}{\left(x+1\right)}+c
=\log\left(x+1\right)-\dfrac{1}{x+1}+c
Hence, the answer is
\log\left(x+1\right)-\dfrac{1}{x+1}+c.
If
\int _{ 0 }^{ \pi /3 }{ \dfrac { \cos { x } }{ 3+4\sin { x } } dx } =k\log { \left( \dfrac { 3+2\sqrt { 3 } }{ 3 } \right) }
, then,
k
is equal to ?
Report Question
0%
\dfrac{1}{2}
0%
\dfrac{1}{3}
0%
\dfrac{1}{4}
0%
\dfrac{1}{8}
Explanation
\sin x=t \Rightarrow dx.\cos x=dt
\displaystyle I=\int_{0}^{\frac{\pi }{3}}\frac{dt}{3+4t}
\displaystyle =\frac{1}{4}[ln(3+4t)]^{\frac{\pi }{3}}
When
x=0\Rightarrow t=0
When
\displaystyle x=\frac{\pi }{3}\Rightarrow t=\frac{\sqrt{3}}{2}
\displaystyle =\frac{1}{4}[ln(3+4t)]^{\frac{\sqrt{3}}{2}}-\frac{1}{4}[ln(3+4t)]^{0}
\displaystyle =\frac{1}{4}ln(3+2\sqrt{3})-\frac{1}{4}ln(3)
\displaystyle I=\frac{1}{4}ln\left[\frac{3+2\sqrt{3}}{3}\right]
K=\dfrac{1}{4}
Find proper substitution
\int _{ 0 }^{ 1 }{ \dfrac { { e }^{ -x } }{ 1+{ e }^{ -x } } dx }
Report Question
0%
1+{ e }^{ -x }\rightarrow t
0%
-{ e }^{ -x }dx\rightarrow dt
0%
-\int _{ 0 }^{ 1 }{ \dfrac { dt }{ t } }
0%
-\int _{ 0 }^{ 1 }{ ln\left| t \right| }
Explanation
\int _{ 0 }^{ 1 }{ \cfrac { { e }^{ -x } }{ 1+{ e }^{ -x } } dx }
1+{ e }^{ -x }=t
\Rightarrow { -e }^{ -x }dx=dt
\int _{ 0 }^{ 1 }{ \cfrac { -dt }{ t } } \quad \therefore 1+{ e }^{ -x }=t
0:0:1
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Practice Class 12 Commerce Maths Quiz Questions and Answers
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