Question

Evaluate $\int \frac{(x+1)(x+\log x{)}^2}{x}dx$

Answer 1

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

Let $x+\log x=t$

Differentiate w.r.t ${}^{\prime }{x}^{\prime }$

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

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

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

Now

$\int \frac{(x+1)(t{)}^2}{x}.\frac{x}{x+1}.dt$

$=\int t^2.dt$

$\frac{t^2+1}{2+1}+c$

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

$=\frac{1}{3}(x+\log x{)}^3+c$