Question

Integrate with respect to $x$:

$\int \frac{1}{x^6{(1+x^{-5})}^{\frac{1}{5}}}dx$

Answer 1

We have,

$I=\int \frac{1}{x^6{(1+x^{-5})}^{\frac{1}{5}}}dx$

Let

$t=1+x^{-5}$

$\frac{dt}{dx}=0-\frac{5}{x^6}$

$\frac{dt}{5}=-\frac{dx}{x^6}$

Therefore,

$I=-\frac{1}{5}\int \frac{1}{t^{\frac{1}{5}}}dt$

$I=-\frac{1}{5}\frac{t^{-\frac{1}{5}+1}}{-\frac{1}{5}+1}+C$

$I=-\frac{1}{5}\frac{t^{\frac{4}{5}}}{\frac{4}{5}}+C$

$I=-\frac{t^{\frac{4}{5}}}{4}+C$

On putting the value of $t$, we get

$I=-\frac{(1+x^{-5}{)}^{\frac{4}{5}}}{4}+C$

Hence, this is the answer.