Mathematics

# $\int { \dfrac { x\sin ^{ -1 }{ x } }{ \sqrt { 1-{ x }^{ 2 } } } } dx =$

##### SOLUTION

Consider the given integral.

$I=\int{\dfrac{x{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}}}dx$

Let $t={{\sin }^{-1}}x$

$dt=\dfrac{dx}{\sqrt{1-{{x}^{2}}}}$

Therefore,

$I=\int{t\sin t}dt$

$I=t\left( -\cos t \right)-\int{1\left( -\cos t \right)dt}$

$I=-t\cos t+\int{\cos tdt}$

$I=-t\cos t+\sin t+C$

On putting the value of $t$, we get

$I=-{{\sin }^{-1}}x\cos \left( {{\sin }^{-1}}x \right)+\sin \left( {{\sin }^{-1}}x \right)+C$

$I=-{{\sin }^{-1}}x\cos \left( {{\sin }^{-1}}x \right)+x+C$

$I=x-{{\sin }^{-1}}x\cos \left( {{\sin }^{-1}}x \right)+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
Evaluate $\int \sin x \, \log (\cos \, x) dx$
• A. $\cos \, x \, \log (\cos \, x)- \cos \, x+ C$
• B. $\cos \, x \, \log (\cos \, x)+ \cos \, x+ C$
• C. none of these
• D. $- \cos \, x \, \log (\cos \, x)+\cos \, x+ C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium

lf $f(\mathrm{x})$ is an even function, and $n\in N$, then $\displaystyle \int_{-\pi}^{\pi}f(x) \sin nx\ dx=$
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$\int {{3^x}{3^{{3^x}}}{3^{{3^{{3^x}}}}}} dx$ is equal to
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The average value of a function f(x) over the interval, [a,b] is the number $\displaystyle \mu =\frac{1}{b-a}\int_{a}^{b}f\left ( x \right )dx$
The square root $\displaystyle \left \{ \frac{1}{b-a}\int_{a}^{b}\left [ f\left ( x \right ) \right ]^{2}dx \right \}^{1/2}$ is called the root mean square of f on [a, b]. The average value of $\displaystyle \mu$ is attained id f is continuous on [a, b].