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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 : 138 of 271
Marks: +1, -0
xlog⁡2xx\log 2x
Solution:  
Let I=∫xlog⁡2x dxI=\int x\log 2x\,dx
=(log⁡2x)⋅x22−∫ddx(log⁡2x)x22 dx=(\log 2x)\cdot\frac{x^2}{2}-\int\frac{d}{dx}(\log 2x)\frac{x^2}{2}\,dx
=log⁡2x⋅x22−∫22xx22 dx+C=\log 2x\cdot\frac{x^2}{2}-\int\frac{2}{2x}\frac{x^2}{2}\,dx+C
=x22log⁡2x−12∫x dx+C=\frac{x^2}{2}\log 2x-\frac{1}{2}\int x\,dx+C
=x22log⁡2x−12⋅x22+C=\frac{x^2}{2}\log 2x-\frac{1}{2}\cdot\frac{x^2}{2}+C
=x22log⁡2x−x24+C=\frac{x^2}{2}\log 2x-\frac{x^2}{4}+C
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