Question Number 195185 by Erico last updated on 26/Jul/23
![1/ Montrer que ∫^( +∞) _( 0) (((1−x^2 )^(2p−1) )/(1−x^(4p) ))dx=((2^(2p−3) /p))π[1+2Σ_(k=1) ^(p−1) cos^(2p−1) (((kπ)/(2p)))] 2/ En de^ duire ∫^( 1) _( 0) (((1−x^2 )^(2p−1) )/(1−x^(4p) ))dx](https://www.tinkutara.com/question/Q195185.png)
$$\mathrm{1}/\:\:\mathrm{Montrer}\:\mathrm{que}\:\underset{\:\mathrm{0}} {\int}^{\:+\infty} \:\frac{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}{p}−\mathrm{1}} }{\mathrm{1}−{x}^{\mathrm{4}{p}} }{dx}=\left(\frac{\mathrm{2}^{\mathrm{2}{p}−\mathrm{3}} }{{p}}\right)\pi\left[\mathrm{1}+\mathrm{2}\underset{{k}=\mathrm{1}} {\overset{{p}−\mathrm{1}} {\sum}}{cos}^{\mathrm{2}{p}−\mathrm{1}} \left(\frac{{k}\pi}{\mathrm{2}{p}}\right)\right] \\ $$$$\mathrm{2}/\:\:\:\:\mathrm{En}\:\mathrm{d}\acute {\mathrm{e}duire}\:\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}{p}−\mathrm{1}} }{\mathrm{1}−{x}^{\mathrm{4}{p}} }{dx} \\ $$