# Integrate the function$\frac{(1+log\: x)^2}{x}$

This question has appeared in model paper 2012

Toolbox:
• (i)Method of substitution :
• Given f(x)dx can be transformed into another form by changing independent variable x to t by substituting x=g(t).
• Consider $I=\int f(x)dx.$
• Put x=g(t) so that $\frac{dx}{dt}=g'(t).$
• $\Rightarrow$dx=g'(t)dt.
• Thus $I=\int f(g(t).g'(t))dt.$
• (ii)$\int \frac{1}{x}dx=log x+c.$
Given $I=\int \frac{(1+log x)^2}{x}dx$.

Put 1+log x=t.

$\frac{1}{x}dx=dt \Rightarrow \frac{1}{x}dx=dt$

Now substituting of x and dx we get,

$I=\int (t)^2.dt.$

On integrating we get,

$\frac{t^3}{3}+c$.

Substituting back for t we get,

$\int \frac{(1+log x)^2}{x}dx=\frac{(1+log x)^3}{3}+c.$