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Question Number 67530 by mathmax by abdo last updated on 28/Aug/19

$$calculate{\int }_0^{\mathrm{\infty }}\frac{x^{n-3}}{1+x^{2n}}dxwithn⩾3$$

Commented by ~ À ® @ 237 ~ last updated on 29/Aug/19

$$letnameditJ$$$$letchangeu=x^{2n}\iff x=u^{\frac{1}{2n}}\Rightarrow dx=\frac{1}{2n}{u}^{\frac{1}{2n}-1}du$$$$J=\frac{1}{2n}.{\int }_0^{\mathrm{\infty }}\frac{u^{\frac{n-3}{2n}}}{1+u}.u^{\frac{1}{2n}-1}du=\frac{1}{2n}.{\int }_0^{\mathrm{\infty }}\frac{u^{\frac{1}{2}-\frac{1}{n}-1}}{1+u}du=\frac{\pi }{2nsin(\frac{\pi }{2}-\frac{\pi }{n})}=\frac{\pi }{2ncos\left(\frac{\pi }{n}\right)}$$

Commented by Abdo msup. last updated on 29/Aug/19

$$thankssir.$$

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