Question

solve: $\int \frac{(1+\log x{)}^2}{x}$

Answer 1

Consider given the expression,

$y=\frac{{(1+\log x)}^2}{x}$

Taking integration both sides,

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

Put,

$t=1+\log x$

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

$dx=x.dt$

$\therefore \int ydx=\int \frac{t^2}{x}x.dt$

$\int ydx=\int t^{2}dt$

$\int ydx=\frac{t^3}{3}+C$

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

Hence, this is the answer.