Question

The value of $\int \frac{(\log x{)}^2}{2x}dx$ is

$\left(a\right)\frac{(\log x{)}^3}{2}+C\left(b\right)\frac{(\log x{)}^3}{3}+C\left(c\right)\frac{(\log x{)}^3}{4}+C\left(d\right)\frac{(\log x{)}^3}{6}+C$

Answer 1

$I=\int \frac{(\log x{)}^2}{2x}dx$
Put $\log x=t\Rightarrow \frac{1}{x}dx=dt$

$\Rightarrow I=\int \frac{t^2}{2}dt$
$=\frac{t^3}{6}+C$
$=\frac{(\log x{)}^3}{6}+C$
Ans : $\left(d\right)$