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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.6
Integrate the functions in Exercises 134 to 155.
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Question : 144 of 271
Marks: +1, -0
xcos⁡−1x1−x2\frac{x\cos^{-1}x}{\sqrt{1-x^2}}
Solution:  
Let I=xcos⁡−1x1−x2dxI=\frac{x\cos^{-1}x}{\sqrt{1-x^2}}dx
Put cos⁡−1x=t\cos^{-1} x = t ⇒−11−x2dx=dt\Rightarrow \frac{-1}{\sqrt{1-x^2}}dx = dt
∴I=−∫tcos⁡tdt\therefore I = -\int t \cos t dt =−[t∫cos⁡tdt−∫(ddtt⋅∫cos⁡t)dt]=-\left[ t\int \cos t dt - \int \left( \frac{d}{dt}t \cdot \int \cos t \right) dt \right]
=−tsin⁡t+∫sin⁡tdt=-t\sin t + \int \sin t dt =−tsin⁡t−cos⁡t+C=-t\sin t - \cos t + C
=−t1−cos⁡2t−cos⁡t+C=-t\sqrt{1-\cos^2 t} - \cos t + C
=−cos⁡−1x1−x2−x+C=-\cos^{-1} x \sqrt{1-x^2} - x + C
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