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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 : 147 of 271
Marks: +1, -0
x(log⁡x)2x (\log x)^2
Solution:  
Let I=∫x(log⁡x)2dx=∫(log⁡x)2⋅xdxI=\int x (\log x)^2 dx=\int (\log x)^2 \cdot x dx
=(log⁡x)2∫xdx−∫(ddx(log⁡x)2⋅∫xdx)dx=(\log x)^2 \int x dx - \int \left( \frac{d}{dx}(\log x)^2 \cdot \int x dx \right) dx
=x22(log⁡x)2−∫(log⁡x)⋅xdx+C=\frac{x^2}{2}(\log x)^2 - \int (\log x) \cdot x dx + C
=x22(log⁡x)2−[(log⁡x)⋅x22−∫1x⋅x22dx]=\frac{x^2}{2}(\log x)^2 - \left[ (\log x) \cdot \frac{x^2}{2} - \int \frac{1}{x} \cdot \frac{x^2}{2} dx \right]
=x22(log⁡x)2−x22log⁡x+12∫xdx=\frac{x^2}{2}(\log x)^2 - \frac{x^2}{2} \log x + \frac{1}{2} \int x dx
=x22(log⁡x)2−x22log⁡x+12⋅x22+C=\frac{x^2}{2}(\log x)^2 - \frac{x^2}{2} \log x + \frac{1}{2} \cdot \frac{x^2}{2} + C
=x22(log⁡x)2−x22log⁡x+x24+C=\frac{x^2}{2}(\log x)^2 - \frac{x^2}{2} \log x + \frac{x^2}{4} + C
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