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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 : 146 of 271
Marks: +1, -0
tan⁡−1x\tan^{-1}x
Solution:  
Let I=∫tan⁡−1xdx=∫tan⁡−1x⋅1dxI=\int\tan^{-1}xdx=\int\tan^{-1}x\cdot 1dx
=tan⁡−1x∫1dx−∫(ddxtan⁡−1x⋅∫1dx)dx=\tan^{-1}x\int 1dx-\int \left( \frac{d}{dx}\tan^{-1}x \cdot \int 1dx \right)dx
=tan⁡−1x⋅x−∫(11+x2)xdx+C=\tan^{-1}x\cdot x-\int \left( \frac{1}{1+x^2} \right)xdx+C
=xtan⁡−1x−12∫2x1+x2dx+C=x\tan^{-1}x-\frac{1}{2}\int \frac{2x}{1+x^2}dx+C
=xtan⁡−1x−12log⁡(1+x2)+C=x\tan^{-1}x-\frac{1}{2}\log(1+x^2)+C
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