Question

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

Answer 1

$\text{we}\text{have}\text{.}$

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

$\text{on}\text{integrate}\text{and}\text{we}\text{get}.$

$\int \frac{(x+1){(x+\text{logx})}^2}{x}dx$

$\text{let}\text{x}\text{+}\text{logx = t1+}\frac{1}{x}=\frac{dt}{dx}$

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

$\text{now,}\int \left(\frac{x+1}{x}\right)t^2\left(\frac{x}{x+1}\right)dt$

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

$=\frac{{\text{t}}^{\text{2+1}}}{\text{2+1}}\text{+ C}$

$\text{=}\frac{{\text{t}}^{\text{2+1}}}{\text{2+1}}\text{+ C}$

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

$\text{=}\frac{\text{1}}{\text{3}}{\text{(x+logx)}}^{\text{3}}\text{+C}$

$\text{Hence,This}\text{is}\text{the}\text{answer}$