Question

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

• A. $\frac{1}{3}{(1+\log x)}^3$
• B. $\frac{1}{2}{(1+\log x)}^3$
• C. $\frac{1}{3}{(1-\log x)}^3$
• D. ${(1+\log x^2)}^3$

Answer 1

The correct option is A $\frac{1}{3}{(1+\log x)}^3$

Let $I=\int \frac{{(1+\log x)}^2}{x}dx$
Put $1+\log x=t\Rightarrow \frac{1}{x}dx=dt$
$I=\int t^{2}dt=\frac{1}{3}{t}^3=\frac{1}{3}{(1+\log x)}^3$
Hence, option 'A' is correct.