$\int \frac{1}{x^{1 / 5}\left(1+x^{4 / 5}\right)^{1 / 2}} d x$, is
Answer & explanation
Correct answer: option 2
Let $I=\int \frac{1}{x^{1 / 5}\left(1+x^{4 / 5}\right)^{1 / 2}} d x$
$\Rightarrow I=\frac{5}{4} \int \frac{1}{\left(1+x^{4 / 5}\right)^{1 / 2}}\left(\frac{4}{5} x^{-1 / 5}\right) d x$
$\Rightarrow I=\frac{5}{4} \int \frac{1}{\left(1+x^{4 / 5}\right)^{1 / 2}} d\left(1+x^{4 / 5}\right)=\frac{5}{2}\left(1+x^{4 / 5}\right)^{1 / 2}+C$