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CBSE Questions for Class 12 Commerce Applied Mathematics Indefinite Integrals Quiz 2 - MCQExams.com
CBSE
Class 12 Commerce Applied Mathematics
Indefinite Integrals
Quiz 2
∫
sin
−
1
(
cos
x
)
d
x
is equal to
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π
x
2
+
c
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π
x
2
2
+
c
0%
π
x
−
x
2
2
+
c
0%
π
x
+
x
2
2
+
c
Explanation
sin
−
1
(
cos
x
)
=
π
2
−
cos
−
1
(
cos
x
)
=
π
2
−
x
∫
sin
−
1
(
cos
x
)
d
x
=
∫
(
π
2
−
x
)
d
x
=
π
x
−
x
2
2
+
C
What is
∫
d
x
x
(
1
+
l
n
x
)
n
equal to
(
n
≠
1
)
?
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1
(
n
−
1
)
(
1
+
l
n
x
)
n
−
1
+
c
0%
1
−
n
(
1
+
l
n
x
)
1
−
n
+
c
0%
n
+
1
(
1
+
l
n
x
)
n
+
1
+
c
0%
−
1
(
n
−
1
)
(
1
+
l
n
x
)
n
−
1
+
c
Explanation
Given,
∫
1
x
(
1
+
ln
(
x
)
)
n
d
x
apply
u
=
1
+
ln
(
x
)
=
∫
1
u
n
d
u
=
∫
u
−
n
d
u
=
u
−
n
+
1
−
n
+
1
=
(
1
+
ln
(
x
)
)
−
n
+
1
−
n
+
1
=
(
1
+
ln
(
x
)
)
−
n
+
1
−
n
+
1
+
C
−
1
(
n
−
1
)
(
1
+
l
n
x
)
n
−
1
+
c
Evaluate
∫
x
s
e
c
x
.
t
a
n
x
d
x
=
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x
sec
x
+
log
|
tan
(
π
/
2
+
x
/
2
)
|
+
c
0%
x
sec
x
−
log
|
tan
(
π
/
4
+
x
/
2
)
|
+
c
0%
x
sec
x
−
log
|
tan
(
π
/
4
+
x
)
|
+
c
0%
x
sec
x
+
log
|
tan
x
/
2
|
+
c
Explanation
∫
x
sec
x
tan
x
d
x
=
∫
x
(
sec
x
(
tan
x
)
)
d
x
=
x
∫
(
sec
x
tan
x
)
d
x
−
∫
1
∫
(
sec
x
tan
x
)
d
x
....... [Using integration parts]
=
x
sec
x
−
∫
sec
x
d
x
=
x
sec
x
−
log
|
sec
x
+
tan
x
|
+
c
=
x
sec
x
−
log
|
tan
(
π
/
4
+
x
/
2
)
|
+
c
.....
[
∵
\therefore \int x\sec x \tan x\ dx=x\sec x-\log\left | \tan (\pi/4+x/2) \right |+c
\displaystyle \int\{\frac{(\log x-1)}{1+(\log x)^{2}}\}^{2}dx
is equal to
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\displaystyle \frac{x}{(\log x)^{2}+1}+c
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\displaystyle \frac{xe^{x}}{1+x^{2}}+c
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^{\dfrac{x}{x^{2}+1}+c}
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\displaystyle \frac{\log x}{(\log x)^{2}+1}+c
Explanation
\int [\dfrac {log x-1}{1+(log x)^2}]^2dx
\int \dfrac {(log x-1)^2}{[1+(log x)^2]^2}=\int \dfrac {(log x)^2+1-2 log x)}{[1+(log x)^2]^2}
=\int (\dfrac {1}{[(1+(log x)^2]}-\dfrac {2 log x}{[1+log^2x]^2})dx
=\int \dfrac {1}{(1+log^2x)}dx-\int \dfrac {2 log x}{(1+log^2x)^2}dx
=\dfrac {1}{(1+log^2x)}\int 1 dx+\int [\dfrac {2 log x}{(1+log^2x)^2}\times\dfrac {1}{x}\int 1\cdot dx]dx-\int \dfrac {2 log x}{(1+log^2x)^2}dx
=\dfrac {x}{1+log^2x}+\int \dfrac {2 log x}{(1+log^2x)^2}dx-\int \dfrac {2 log x}{(1+log^2x)^2}dx+c
=\dfrac {x}{1+log^2x}+c
\displaystyle \int \sec^{2}x.\text{cosec}^{2}xdx=
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\tan x-\cot x +c
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\tan x +\cot x +c
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-\tan x + \cot x +c
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\sec x \tan x +c
Explanation
\displaystyle \int \sec^{2}x. \text{cosec}^{2}xdx
\displaystyle =\int \dfrac{1}{\cos^{2}x.\sin^{2}x}dx
\displaystyle =\int \dfrac{\cos^{2}x+\sin^{2}x}{\cos^{2}x.\sin^{2}x}dx
\displaystyle =\int \dfrac{\cos^{2}x}{\cos^{2}x\sin^{2}x}+\dfrac{\sin^{2}x}{\cos^{2}x.\sin^{2}x}dx
\displaystyle =\int \dfrac{1}{\sin^{2}x}+\dfrac{1}{\cos^{2}x}dx
\displaystyle =\int (\text{cosec}^{2}x+\sec^{2}x) dx
=-\cot x+\tan x +c
=\tan x-\cot x+c
\displaystyle \int\sqrt{\frac{x}{x-1}}dx, x\in(0,\pi/2)
equals
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\sqrt{x(x-1)}+\log(\sqrt{x}+\sqrt{x-1})+c
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\sqrt{x(x-1)}-\log(\sqrt{x}+\sqrt{x-1})+c
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\sqrt{x(x-1)}+\log(\sqrt{x}-\sqrt{x-1})+c
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\sqrt{x(x+1)}-\log(\sqrt{x}-\sqrt{x+1})+c
Explanation
x=sec^{2}\theta
dx =2 sec^{2}\theta \cdot tan \theta \cdot d\theta
\int \sqrt{\dfrac{sec^{2}\theta}{tan^{2}\theta}} \cdot2 sec^{2}\theta \ tan\ \theta d\theta
\int \dfrac{1}{sin \theta} \cdot 2 sec^{2}\theta \cdot \dfrac{sin \theta}{cos \theta} \cdot d\theta
=2\int sec^{3}\theta \cdot d\theta \cdot
=2\int (sec \theta)(sec^{2}\theta)d\theta \cdot
=2[sec\theta tan \theta -\int (sec \theta tan \theta \cdot tan \theta)d\theta \cdot ]
2\int sec^{3}\theta ^{d\theta}=2[sec\theta tan \theta]-2\int sec \theta(sec^{2}\theta-1)d\theta
2\int sec^{3}\theta d\theta =sec\theta tan \theta +\int sec \theta d\theta
2\int sec^{3} \theta d\theta = sec\theta tan\theta \rightarrow log |sec\theta+tan \theta |
and
=\sqrt{x(x-1)}+log |\sqrt{x}+\sqrt{x-1}|+c
\int \dfrac{x}{\sqrt{x-1}} dx = \sqrt{x(x-1)} +log |\sqrt{x}+\sqrt{x-1}|+c \cdot
\displaystyle \int xe^{2x}(1+x)dx
equal to
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\displaystyle \frac{xe^{x}}{2}+c
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\displaystyle \frac{(e^{x})^{2}}{2}r
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\displaystyle \frac{(1+x)^{2}}{2}+c
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\displaystyle \frac{(xe^x)^2}{2}
Explanation
\displaystyle I=\int xe^{2x}(1+x)dx
\displaystyle =\int xe^{2x}dx+\int x^{2}e^{2x}dx
\displaystyle =e^{2x} \frac{x^{2}}{2}-\int x^{2}e^{2x}dx+\int x^{2}e^{2x}dx
\displaystyle I=\frac{(xe^{x})^{2}}{2}
The value of
\displaystyle \int x^{2}2^{3x}dx=
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\displaystyle \frac{x^{2}2^{3x}}{4\log 2}-\frac{x.2^{3x+1}}{9.(\log 2)^3}+\frac{2^{3x+1}}{27(\log 2)}+c
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\displaystyle \frac{x^{2}2^{3x}}{3\log 2}-\frac{x.2^{3x+1}}{9.(\log 2)}+\frac{2^{3x+1}}{27(\log 2)}+c
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\displaystyle \frac{x^{2}2^{3x}}{3\log 2}-\frac{2^{3x+1}.x}{3^{2}(\log 2)^{2}}+\frac{2^{3x+1}}{3^{3}(\log 2)^{3}}+c
0%
\displaystyle \frac{x^{2}2^{3x}}{\log 2}-\frac{2^{3x+1}.x}{3^{2}(\log 2)^{2}}+\frac{2^{3x+1}}{(\log 2)^{3}}+c
Explanation
\int x^2\cdot 2^{3x} dx\cdot
=x^2\int 2^{3x}dx+\int 2x\int 2^{3x}\cdot dx\cdot
=x^2\dfrac {2^{3x}}{log 8}-\int 2x\dfrac {2^{3x}}{log 8^{dx}}
=\dfrac {x^22^{3x}}{log 8}-\dfrac {2}{log 8}[\dfrac {x2^{3x.}}{log 8}-\int \dfrac {2^{3x}}{log 8^{.dx}}]
=\dfrac {x^22^{3x}}{log 8}-\dfrac {2}{log 8}[\dfrac {x2^{3x}}{log 8}-\dfrac {1}{log 8}\dfrac {2^{3x}}{log 8}]+c.
=\dfrac {x^{2}2^{3x}}{log 8}-\dfrac {2^{3x+1}\cdot x}{3^2(log 2)^2}+\dfrac {2^{3x+1}}{3^3(log 2)^{3}}+c
\displaystyle \int\log{x}d{x}=
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x(\log x-x)+c
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x(\log x)+c
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x(\log x-1)+c
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x(\log x+x)+c
Explanation
\int\ log\ x\ dx= \int\ log\ x-1\ dx
= log\ x\int\ 1-dx-\ \int(^1/_x.\sqrt{1.dx})dx
= log\ x(x)-x+c
= x(log\ x-1)+c
\because \int \ log\ x\ dx= x(log\ x-1)+c
Evaluate :
\displaystyle \int\frac{\cot^{2}x}{(co\sec^{2}x+co\sec x)}d{x}
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x-\sin x +C
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x + \cos x +C
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\sin x - x+C
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2\tan(\displaystyle \frac{ax}{2})+c
Explanation
\displaystyle \int\frac{\cot^{2}x}{(co\sec^{2}x+co\sec x)}d{x}
\displaystyle \int \dfrac{cos ec^{2}x-1}{cos ecx(cosecx+1)}dx
\displaystyle \int \dfrac{(cosecx-1)(cosecx+1)}{cos ecx(cosecx+1)}dx
\displaystyle \int \dfrac{cosecx-1}{cosecx}dx=\int (1-sin x)dx
=x+cos x+c
\because -\int sin x.dx=cos x+c
\displaystyle \int[\frac{cosx}{x}-\sin{x}\log x]dx=
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(\log x) sinx+c
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(\log x) (cos x) +c
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2(\log x) (cos x) +c
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-(\log x) sin x+c
Explanation
\int\left [ \displaystyle \dfrac{cos\ x}{x}-sinx\ log\ x \right ]dx
\int \displaystyle \dfrac{cos\ x}{x}dx-\ \int\ sin\ x\ log\ x\ dx
cos\ x\ \int \displaystyle \dfrac{1}{x}\ dx+\ \int\ sin\ x\ log\ x.\ dx
-\int\ sin\ x\ log\ x\ dx.
= (\ cos\ x)( log\ x)+c
Hence,
\int \left [ \displaystyle \dfrac{cos x}{x}-(sin\ x)(log\ x) \right ]dx=(\ cos\ x)(\ log\ x)+c
Evaluate
\displaystyle \int x^{2}e^{x}dx=
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e^{x}(x^{2}-2x+2)+c
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e^{x}(x^{2}+2x+2)+c
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x^{2}+ex+c
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e^{x}(x^{2}+x+2)+c
Explanation
Let
I=\displaystyle \int x^{2}e^{x} dx
= x^{2} \displaystyle \int e^{x}dx
-\displaystyle \int \left (\frac{dx^{2}}{dx}\cdot \int e^{x} dx\right).dx
=x^{2} e^{x} -\displaystyle \int (2xe^{x}) dx\cdot
=x^{2}e^{x}-2\displaystyle \int xe^{x}\cdot dx\cdot
Taking
\displaystyle \int xe^{x}= xe^{x}-e^{x}\cdot
....(by using by parts)
Therefore,
I=x^{2}e^{x}-2[xe^{x}-e^{x}]+c
=e^{x}[x^{2}-2x+2]+c
\displaystyle \int e^{\log x}\cos xdx=
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xsinx-cosx+c
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\dfrac{x}{2}sinx+cos^{2}x+c
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xsinx+cosx+c
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xsinx+sinx+c
Explanation
\int e^{logx}cosx\ dx=\int x\ cosx\ dx.
=x\int cosx\ dx-\ \int \left ( 1\int cosxdx \right )dx
=x\ sinx+cosx+c
\int e^{logx}cosx\ dx=\ x\ sinx+cosx+c.
\displaystyle \int\frac{x\cos x\log x-\sin x}{x(\log x)^{2}}dx
is equal to
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\displaystyle \frac{\sin x}{\log x}+c
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\log{x}+\sin{x+c}
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\log{x}\sin{x+c}
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\displaystyle \frac{\cos x}{\log x}+c
Explanation
\displaystyle\int\frac{x\cos x\log\ x-\sin x}{x(\log\ x)^{2}}dx
=\displaystyle\int \frac{\cos x}{\log x} dx - \int\frac{\sin x}{x(\log\ x)^{2}}dx
Using Integration by parts in the first expression, we get
=\displaystyle\frac{1}{\log\ x}\int\cos x dx -\int\left(\frac{d\left (\dfrac{1}{\log x}\right )}{dx}\cdot\int\cos x dx\right)dx-\int \frac{\sin x}{x(\log x)^{2}}dx
=\displaystyle\frac{1}{\log x}(\sin x)+\int\frac{1}{x(\log x)^{2}}\sin x dx-\int \frac{\sin x}{x(\log x)^{2}}dx
=\dfrac{\sin x}{\log x}+c
where,
c
is a constant of integration.
The value of
\int x(cosecx\ cotx)dx=
is
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xcosec x-log |tanx/2|+c
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2-xcosec x+\log| tan \displaystyle \frac{x}{2}|+c
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xcosec x-2 \log|tanx/2|+c
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xcot x-log |tan{\displaystyle \frac{x}{2}}|+c
Explanation
\int x(cosecx\ cot)dx=x\int (cosecx\ cotx)dx
-\int 1\left ( \int cosecx\ cot\ xdx \right )dx
=x\left [ -cosecx \right ]+c+\int (cosecx)dx
=\ -x\ cosecx+c+log\left | cosecx-cotx \right |
=-x\ cosecx+c+\ log\left | \dfrac{1-cosx}{sinx} \right |
=2-x\ cosecx+c_{1}+log\left | tan\ x/2 \right |
\int x\ cosecx\ cotx\ dx
=2-x\ cosecx+c_{1}+log\ \left | tan\ \dfrac{x}{2} \right |
Solve
\displaystyle \int \tan^{-1}xdx
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x\tan x -\log|1+x^{2}|+c
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x
\tan^{-1} x -\displaystyle \frac{1}{2}\log|1+x^{2}|+c
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x
\tan^{-1} x+\log\sqrt{1+x^{2}}+c
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x\tan x+\log|1+x^{2}|+c
Explanation
\displaystyle \int \tan^{-1}x \ .1 \ \ dx
Let us assume here
\tan^{-1}x
is the first function and constant 1 is the second function. Then the integral of the second function is x.
\displaystyle \int u.v \ dx=u\int v\ dx
\displaystyle -\int \left ( \dfrac{du}{dx}\int v\ dx \right )dx
Now by integrating the functions by parts, we get
\displaystyle =\tan^{-1}x\int 1\ dx
\displaystyle -\int \left ( d\frac{\ tan^{-1}x}{dx}\int 1\ dx \right )dx
\displaystyle =x\tan^{-1}x-\int \frac{1}{1+x^{2}} x\ dx
\displaystyle =x\tan^{-1}x+c-\frac{1}{2}\int \frac{2x\ dx}{1+x^{2}}
\displaystyle =-\ x\tan^{-1}x-\frac{1}{2}\log(1+x^{2})+c
=\displaystyle \int \tan^{-1}x\ dx=x\tan^{-1}x-\frac{1}{2}\log(1+x^{2})+c
Evaluate
\displaystyle \int\frac{\log(x/e)}{(\log x)^{2}}dx
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\displaystyle \frac{\log x}{x}+c
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\displaystyle \frac{x}{\log x}+c
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^{\dfrac{x}{log(x)^{2}}+c}
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\displaystyle \frac{(\log x)^{2}}{x}+c
Explanation
\displaystyle \int \dfrac{\log (x/e)}{(\log x)^{2}}dx=\int\ \dfrac{\log \ x-1}{(\log x)^{2}}
=\displaystyle \int\ \left [ \dfrac{1}{(\log \ x)}-\dfrac{1}{(\log x)^{2}} \right ]dx
=\displaystyle \int\ \dfrac{1}{(\log \ x)}dx-\ \int\ \dfrac{1}{(\log \ x)^{2}}dx
=\displaystyle \dfrac{1}{(\log \ x)}\int\ 1.dx+\ \int\ \dfrac{1}{(\log \ x)^{2}}dx-\int\ \dfrac{1}{(\log \ x)^{2}}dx
..... [Using integration by parts]
=\ \dfrac{x}{\log \ x}+c
\therefore \displaystyle \int\ \dfrac{\log {(x/e)}}{(\log \ x)^{2}}dx=\dfrac{x}{\log x}+c
\displaystyle \int x^{2} {\it cosh} 4xdx=
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\displaystyle \frac{x^{2}}{4} {\it sinh}4x-\displaystyle \frac{x}{8}{\it cosh} 4x+\displaystyle \frac{\sinh 4x}{32}+c
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\displaystyle \frac{x^{2}}{4}sinh4x+\frac{x}{8}
cosh
4{x-}\displaystyle \frac{\sinh 4x}{32}+c
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\displaystyle \frac{x^{2}}{4}sinh4x-\frac{x}{8}
cosh
4x-\displaystyle \frac{\sinh 4x}{32}+c
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\displaystyle \frac{x^{2}}{4}sinh4x+\frac{x}{8}
cosh
4x-\displaystyle \frac{\sin h4x}{64}+
Explanation
\int x^{2}\ cosh\ 4x.dx
x^{2}\int cosh\ 4x\ dx-\int 2x\left [ \left ( \int cosh\ 4x \right )dx \right ]dx
=\dfrac{x^{2}}{4}sin\ h4x-\ 1/2\ \int (x\ sinh4x)dx\ (1)
\int x\ sinh\ 4x\ dx=\dfrac{x}{4}\ cos\ h4x-\int \dfrac{cosh}{4}4x\ dx
=\left ( \dfrac{x}{4}cos\ h\ 4x-\dfrac{sin}{16} 4x\right )\ -(2)
Using (2) in (1),
=\dfrac{x^{2}}{4}sin\ h4x-\dfrac{x}{8}cosh\ 4x+\dfrac{sinh4x}{32}+c
\int x^{2}\ cosh\ 4x\ dx=\dfrac{x^{2}}{4}sin\ h4x-x/8\ cosh\ 4x+\dfrac{sinh\ 4x}{32}+c
\displaystyle \int cot^{-1}(\frac{x-1}{x+1})dx=
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\displaystyle \frac{\pi}{4}x+x\cot^{-1}x-\frac{1}{2}\log(1+x^{2})+c
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x\displaystyle \cot^{-1}x+\frac{1}{2}\log(1+x^{2})+c
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\displaystyle \frac{\pi}{4}x-\frac{1}{2}\log(1+x^{2})+c
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^{\displaystyle \frac{\pi}{4}x+x\cot^{-1}x}+\displaystyle \frac{1}{2}\log(1+x^{2})+c
Explanation
cosec\theta =\sqrt{1+x^{2}}
sin \theta =\dfrac{1}{\sqrt{1+x^{2}}}
\int cot^{-1}(\dfrac{x-1}{x+1})dx
x=cot\theta
\int cot^{-1}(\dfrac{cot\theta -1}{cot\theta +1})[cosec^{2}\theta ]d\theta
-\int cot^{-1}[cot(\pi /4+\theta )](cosec^{2}\theta )d\theta
=-\int (\pi /4+\theta (cosec^{2}\theta ))d\theta
=-[\int \pi /4cosec^{\gamma }\theta d\theta +\int \theta cosec^{\gamma \theta d\theta }]
=-[-\pi /4cot\theta +[\theta (-cot\theta )+\int cot\theta d\theta ]
=\pi /4cot\theta +cot\theta -log|sin\theta |+c
=\pi /4x+x cot^{-1}x-1/2log(\sqrt{1+x^{2}})+c
\int cot^{-1}(\dfrac{x-1}{x+1})dx=\pi /4{x}+x cot^{-1}x -1/2 log (\sqrt{1+x^{2}})+c
\displaystyle \int e^{x}cosec x(1-\cot x)dx=
Report Question
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e^{x}\cot x+c
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e^{x}cosec x\cot x+c
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e^{x}cosec x+c
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-e^{x}\cot x+c
Explanation
\int e^{x}cosec\ x(1-cot\ x)dx
\int e^{x}\left ( cosec\ x-cose\ x\ cot\ x \right )dx
\int e^{x}cosec\ x\ dx-\int e^{x}\left ( cosec\ x.cot\ x \right )dx
\int e^{x}\left ( cosec\ x\ cot\ x \right )dx=ex\int cosec\ x\ cot\ x\ dx+\int e^{x}cosec\ x.dx
=e^{x}cosec\ x+\int cosec\ x\ dx.
\int e^{x}cosec\ x\ dx-\left ( -e^{x}cosec\ x+\int cosec\ x\ dx \right )+c
=\int e^{x}cosec\ x\ dx+e^{x}\ cosec\ x-\int e^{x}\ cosec\ x\ dx
=\ e^{x}cosec\ x+c
\displaystyle \int x^{2}a^{x}dx=
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a^{x}[\displaystyle \frac{x^{2}}{\log a}-\frac{2x}{(\log a)^{2}}+\frac{2}{(\log a)^{3}}]+c
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a^{x}[\displaystyle \frac{x^{2}}{\log a}+\frac{2x}{(\log a)^{2}}-\frac{2}{(\log a)^{3}}]+c
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a^{x}[\displaystyle \frac{x^{2}}{\log a}+\frac{2x}{(\log a)^{2}}+\frac{2}{(\log a)^{3}}]+c
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a^{x}(\displaystyle \frac{x}{\log a}-\frac{1}{(\log a)^{2}})+c
Explanation
\int x^{2}a^{x}dx = x^{2}\int a^{x}dx-\int \left [ \dfrac{dx^{2}}{dx}\int a^{x}\ dx \right ]dx
\dfrac{x^{2}a^{x}}{log\ a}-\int \left [ 2x\int a^{x}dx \right ]dx
\dfrac{x^{2}a^{x}}{log\ a}-2\int \dfrac{x\ a^{x}}{log^{a}}dx
\dfrac{x^{2}a^{x}}{log\ a}-\dfrac{2}{log^{a}}\int [xa^{x}]dx
\int xa^{x}dx=\dfrac{xa^{a}}{ ^{a}}-\int 1\dfrac{a^{x}}{(log^{a})}dx
=\dfrac{xa^{x}}{log^{a}}-\dfrac{a^{x}}{(log^{a})^{2}}
=\dfrac{x^{2}a^{x}}{log^{a}}-\dfrac{2}{log^{a}}\left [ \dfrac{xa^{x}}{log^{a}}-\dfrac{a^{x}}{(log^{a})^{2}} \right ]
=\dfrac{x^{2}a^{x}}{log^{a}}-\dfrac{2}{log^{a}}\left [ \dfrac{xa^{x}}{log^{a}}-\dfrac{a^{x}}{(log^{a})^{2}} \right ]
a^{x}\left [ \dfrac{x^{2}}{log{a}}-\dfrac{2x}{(loga)^{2}}+\dfrac{2}{(log\ a)^{3}} \right ]+c
lf polynomials
P
and
Q
satisfy
\displaystyle{\int((3x-1)\cos x+(1-2x)\sin x)dv}=P\cos x+Q\sin x+R
(ignoring the constant of integration) then
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P=3x-2
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Q=2+x
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P=3(x-1)
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Q=3(x-1)
Explanation
Let
I=\int((3x-1)\cos x+(1-2x)\sin x)dx
Using integration by parts, we evaluate the 2 expressions individually,
\int (3x-1) \cos x = (3x - 1) \sin x - \int 3\sin x dx
= (3x - 1) \sin x + 3 \cos x + c
Similarly,
\int (1-2x)\sin x dx = -(1-2x)\cos x - \int (-2)(-\cos x) dx
= (2x - 1) \cos x - 2\sin x + c
Adding both the expressions, we get
I= 3(x - 1)\sin x + 2(x+1)\cos x
Ignoring the constant
Comparing it with the RHS, we get
P = 2(x+1)
and
Q = 3(x-1)
Hence, option 'D' is correct.
Evaluate
\displaystyle \int\frac{\arcsin\sqrt{x}}{\sqrt{1-x}}dx
=
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2[\sqrt{x}-\sqrt{1-x}
arc
\sin\sqrt{x}]+c
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2[\sqrt{x}+\sqrt{1-x}
arc
\sin\sqrt{x}]+c
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2[\sqrt{x}+\sqrt{1-x}
arc
\cos\sqrt{x}]+c
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2[\sqrt{x}-\sqrt{1-x}
arc
\cos\sqrt{x}]+c
Explanation
\displaystyle \int \dfrac {\sin ^{-1}\sqrt x}{\sqrt {1-x}}\cdot dx.
Put
x=\sin ^2\theta
\Rightarrow dx=2 \sin \theta \cos \theta d\theta.
\therefore \displaystyle \int \dfrac {\sin ^{-1}\sqrt x}{\sqrt {1-x}}\cdot dx.
=\displaystyle \int \dfrac {\sin ^{-1}(\sin \theta)}{\sqrt {1-\sin ^2\theta}}\cdot d\theta(2 \sin \theta)(\cos \theta)
\displaystyle =2\int \theta \sin \theta d\theta
=2[\theta(-\cos \theta)+\int \cos \theta d\theta.]
=2[\theta(-\cos \theta)+\sin \theta]+c
=2[\sin \theta-\theta \cos \theta]+c
=2[\sqrt x-(\sin ^{-1}\sqrt x)\sqrt {1-x}]+c
\therefore \displaystyle \int \dfrac {\sin ^{-1}\sqrt x}{\sqrt {1-x}}dx=2[\sqrt x-(\sqrt {1-x})\sin ^{-1}\sqrt x]+c
\displaystyle \int e^{x}(x^{2}-5x+8)
dx
=e^{x}f(x)+c
then
f(x)
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x^{2}-5x+12
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x^{2}+7x+15
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x^{2}-7x-15
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x^{2}-7x+15
Explanation
\int e^{x}(x^{2}-5x+8)dx
=\int e^{x}x^{2}dx-5\int e^{x}.xdx+8\int e^{x}\ dx
=x^{2}e^{x}=2\int xe^{x}\ dx-5\int e^{x}\ x+8\int e^{x}dx
x^{2}e^{x}-7[xe^{x}-e^{x}]+8e^{x}
=x^{2}e^{x}-7xe^{x}+15\ e\ ^{x}
=e^{n}[x^{2}-7x+15]+c
so,\ f(x)-\ x^{2}-7x+15
Evaluate
\displaystyle \int x\frac{(\sec 2x-1)}{(\sec 2x+1)}dx
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x \tan x-\log |\displaystyle \sec x|+\frac{x^{2}}{2}+c
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x \tan x-\log |\displaystyle \sec x|-\frac{x^{2}}{2}+c
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x \tan x-\log |\sec\frac{x}{2}|+c
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x \tan x-\log |\displaystyle \sec\frac{x}{2}|+\frac{x^{2}}{2}+c
Explanation
\displaystyle \int x\left(\dfrac{\sec 2x-1}{\sec 2x+1}\right)dx=\int x\left(\dfrac{1-\cos 2x}{1+\cos 2x}\right)dx
=\displaystyle \int x \tan^{2}x dx
\displaystyle \int x(\sec ^{2}x-1)dx
\displaystyle =\int x \sec ^{2}x \ {dx}-\int xdx
We know,
\displaystyle \int u.v \ dx=u\int v\ dx
\displaystyle -\int \left ( \dfrac{du}{dx}\int v\ dx \right )dx
Now by integrating the functions by parts, for the first part taking
x
as
u
and
sec^2 x
as
v
.
\displaystyle =x\int \sec ^{2}xdx-\int 1\left(\int \sec ^{2}xdx\right)dx-\int xdx
\displaystyle =x \tan x-\int \tan x \ dx-\dfrac{x^{2}}{2}+c
=x \tan x-\log |\sec x|-\dfrac{x^{2}}{2}+c
\displaystyle \int x\left(\dfrac{\sec 2x-1}{\sec 2x+1}\right)dx=x \tan x-\log|\sec x|-\dfrac{x^{2}}{2}+c
\displaystyle \int e^{x}2^{3\log_{2}x}dx=e^{x}f(x)+c
, then
f(x)=
Report Question
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{ x }^{ 3 }-3{ x }^{ 2 }+6x-6
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x^{3}-3x^{2}-6x-3
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x^{3}-3x^{2}+6x+6
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x^{3}+3x^{2}+6x+6
Explanation
Given,
\displaystyle \int e^{x}2^{3\log_{2}x}dx=e^{x}f(x)+c
......(A)
Consider,
\int e^{ x }2^{ 3\log _{ 2 }{ x } }dx=\int e^{ x }.x^{ 3 }dx
\int e^{x} \times x^{3}dx=x^{3}\int e^{x}dx-\int 3x^{2}e^x\ dx
=x^{3}e^{x}-3 \int x^{2}e^{x}\ dx\
....(1)
\int x^{2}e^{x}=x^{2}e^{x}-2\int xe^{x}\ dx\
.....(2)
\int xe^{x}=\ xe^{x}-e^{x}\
.....(3)
Using (3) in (2),
\int x^{2}e^{x}=x^{2}e^{x}-2(xe^{x}-e^{x})\
....(4)
Using (4) in (1)
\displaystyle \int e^{ x }.x^{ 3 }dx= x^{3}e^{x}-3\left [ x^{2}e^{x}-2(xe^{x}-e^{x}) \right ]+c
=x^{3}e^{x}-3x^{2}e^{x}+6xe^{x}-6e^{x}+c
=e^{x}\left [ x^{3}-3x^{2}+6x-6 \right ]+c
On comparing with eqn (A), we get
f(x)=x^{3}-3x^{2}+6x-6
\displaystyle \int\frac{x+\sin x}{1+\cos x}dx=
Report Question
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xtan \displaystyle \frac{x}{2}+c
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xcot \displaystyle \frac{x}{2}+c
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xsin \displaystyle \frac{x}{2}+c
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xcos \displaystyle \frac{x}{2}+c
Explanation
\displaystyle \int\frac{x+\sin x}{1+\cos x}dx
\displaystyle =\frac { 1 }{ 2 } \int \frac { x+\sin x }{ \cos ^{ 2 }{ (x/2 )} } dx
\displaystyle =\frac { 1 }{ 2 } \int x\sec ^{ 2 }{ (x/2) } dx+\int \tan { (x/2) } dx
\displaystyle=\frac { 1 }{ 2 } [x\frac { \tan { (x/2) } }{ 1/2 } -\int \frac { \tan { (x/2) } }{ 1/2 } dx]+\int \tan { (x/2) } dx+C
=x\tan { (x/2) } +C
\displaystyle \int\cos^{-1}(\frac{1}{x})dx
equal to
Report Question
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x\sec^{-1}x+\cosh^{-1}x+c
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x\sec^{-1}x-\cosh^{-1}x+c
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x\sec^{-1}x-\sin^{-1}x+c
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x\sec^{-1}x+\sin^{-1}x+c
Explanation
x=sec\theta
dx=sec\theta \cdot tan\theta d\theta.
\int cox^{-1}(cos\theta)\cdot (sec\theta \cdot tan\theta)d\theta.
\int \theta (sec\theta \cdot tan\theta)d\theta \cdot \theta \int (sec\theta \cdot tan\theta)d\theta -\int (\int(sec\theta \cdot tan\theta)d\theta)d\theta
\int sec\theta tan\theta\cdot d\theta=sec\theta+c
=\theta sec\theta-\int sec\theta d\theta
=\theta sec\theta+(log \mid sec\theta+tan\theta \mid)+1.
=x sec^{-1}x.-log \mid x+\sqrt {x^2-1}\mid +c
=x sec^{-1}x-cos h^{-1}x+c.
\int cos^{-1}(\dfrac {1}{x})=x sec^{-1}{x}-cos h^{-1}x+c.
Evaluate:
\displaystyle \int\log_{10}xdx
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(x-1)\log_{e}x+c
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\displaystyle \log_{e}10.x\log_{e}(\frac{x}{e})+c
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\displaystyle \log_{10}e.x\log_{e}(\frac{x}{e})+c
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\cfrac{1}{x}+c
Explanation
We know that,
\int { uvdx } =u'\int { vdx } -\int { u'(\int { vdx } )dx } +C
(Integration by parts rule)
\therefore
we can write,
I=\int { \log _{ 10 }{ x } dx } =\log _{ 10 }{ e } \int { \log _{ e }{ x } \cdot (1)dx }
...as
\log _{ 10 }{ x } =\cfrac { \log _{ e }{ x } }{ \log _{ e }{ 10 } } =\log _{ 10 }{ e } \cdot \log _{ e }{ x }
Applying integration by parts;
u=\log _{ e }{ x } v=1\\ u'=\cfrac { 1 }{ x } \int { vdx } =x\\
\therefore I=\log _{ 10 }{ e } ((\log _{ e }{ x } )x-\int { \cfrac { 1 }{ x } \cdot xdx } )+C=\log _{ 10 }{ e } (x\log _{ e }{ x } -\int { dx } )+C\\ I=\log _{ 10 }{ e } \cdot x(\log _{ e }{ x } -1)+C=\log _{ 10 }{ e } \cdot x(\log _{ e }{ x } -\log _{ e }{ e } )+C\\ \therefore I=\log _{ 10 }{ e } \cdot x(\log _{ e }{ \cfrac { x }{ e } } )+C
Evaluate
\displaystyle \int\frac{xe^{x}}{(x+1)^{2}}dx
Report Question
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\displaystyle e^{x}[\dfrac{1}{x+1}]+c
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\displaystyle e^{x}[\dfrac{-1}{x+1}]+c
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e^{x}x+c
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\displaystyle e^{x}[\dfrac{-1}{(x+1)^{2}}]+c
Explanation
I = \displaystyle \int\frac{xe^{x}}{(x+1)^{2}}dx
\displaystyle =\int\frac{(x+1-1)e^{x}}{(x+1)^{2}}dx
\displaystyle =\int \frac{(x+1)e^{x}}{(x+1)^{2}}dx -\int\frac{e^{x}}{(x+1)^{2}}dx
\displaystyle =\int \frac{e^{x}}{x+1} dx-\int\frac{e^{x}}{(x+1)^{2}}dx
We know,
\displaystyle \int u.v \ dx=u\int v\ dx
\displaystyle -\int \left ( \dfrac{du}{dx}\int v\ dx \right )dx
Now by integrating the functions by parts, for the first part taking
e^x
as
u
and
\dfrac{1}{x+1}
as
v
I\displaystyle =\frac{e^{x}}{x+1} +\int\frac{e^{x}}{(x+1)^{2}}dx-\int\frac{e^{x}}{(x+1)^{2}}dx
\displaystyle I=\frac{e^{x}}{x+1}+C
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers
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