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Q36  Integrate the functions  \frac{( x+1)( x+ \log x )^2}{x }

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Given function  \frac{( x+1)( x+ \log x )^2}{x }

Simplifying to solve easier;

\frac{( x+1)( x+ \log x )^2}{x } = \left ( \frac{x+1}{x} \right )\left ( x+\log x \right )^2

                                         =\left ( 1+\frac{1}{x} \right )\left ( x+\log x \right )^2

Assume that x+\log x =t

\therefore \left ( 1+\frac{1}{x} \right )dx = dt

= \int \left ( 1+\frac{1}{x} \right )\left ( x+\log x \right )^2 dx = \int t^2 dt

= \frac{t^3}{3}+C

Now, back substituting the value of t,

= \frac{(x+\log x )^3}{3}+C

 

Posted by

Divya Prakash Singh

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