2
$\begingroup$

Calculate using Euler integrals $$\int \limits_{-1}^{1} \frac{dx}{\sqrt[3]{(1+x)^2(1-x)}}$$

What substitution is better to use here?

I have tried something like $t = \frac{1-x}{1+x}$ but it doesn't look good.

$\endgroup$
5
$\begingroup$

If you set $x=2t-1$ you get

$$ 2\int_{0}^{1}\frac{dt}{\sqrt[3]{(2t)^2(2-2t)}}=\int_{0}^{1} t^{-2/3}(1-t)^{-1/3}\,dt = \frac{\Gamma(1/3)\,\Gamma(2/3)}{\Gamma(1)} = \frac{\pi}{\sin\frac{\pi}{3}}=\color{red}{\frac{2\pi}{\sqrt{3}}}.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.