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Q 14  Integrate the functions x ( \log x )^ 2

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Consider x ( \log x )^ 2

So, we have then: I = \int x(\log x)^2 dx

After taking (\log x )^2 as a first function and x as second function and integrating by parts, we get

I = (\log x )^2 \int xdx -\int \left \{ \left ( \frac{d}{dx} (\log x)^2 \right )\int x.dx \right \}dx

= (\log x)^2 .\frac{x^2}{2} - \int \frac{2\log x }{x}.\frac{x^2}{2} dx

= (\log x)^2 .\frac{x^2}{2} - \int x\log x dx

= (\log x)^2 .\frac{x^2}{2} - \left ( \frac{x^2 \log x }{2} -\frac{x^2}{4} \right )+C

 

Posted by

Divya Prakash Singh

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