Question

f(x) = $\int \frac{dx}{x^2{(1+x^5)}^{4/5}}$, taking c = 0, value of $|f\left(1\right)|=$

Answer 1

$I=\int \frac{dx}{x^2{(1+x^5)}^{4/5}}\int \frac{dx}{x^6{\left(\frac{1}{x^5+1}\right)}^{4/5}}$
$Lett=1+\frac{1}{x^5}ordt=\frac{5dx}{x^6}$
$\therefore I=-\frac{1}{5}\int \frac{dt}{t^{4/5}}=-t^{3/5}+c=-{(1+\frac{1}{x^5})}^{3/5}+c$
$=-{\left(\frac{1+x^5}{x}\right)}^{\frac{3}{5}}+c$
$f\left(x\right)=-{\left(\frac{1+x^5}{x}\right)}^{\frac{3}{5}}\Rightarrow f\left(1\right)=-2\Rightarrow |f\left(1\right)|=2$