calculate-0-x-n-3-1-x-2n-dx-with-n-3-

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

c a l c u l a t e \int_{0}^{\infty} \frac{x^{n - 3}}{1 + x^{2 n}} d x w i t h n ⩾ 3

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

l e t n a m e d i t J

l e t c h a n g e u = x^{2 n} \Leftrightarrow x = u^{\frac{1}{2 n}} \Rightarrow d x = \frac{1}{2 n} u^{\frac{1}{2 n} - 1} d u

J = \frac{1}{2 n} . \int_{0}^{\infty} \frac{u^{\frac{n - 3}{2 n}}}{1 + u} . u^{\frac{1}{2 n} - 1} d u = \frac{1}{2 n} . \int_{0}^{\infty} \frac{u^{\frac{1}{2} - \frac{1}{n} - 1}}{1 + u} d u = \frac{π}{2 n s i n (\frac{π}{2} - \frac{π}{n})} = \frac{π}{2 n c o s (\frac{π}{n})}

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

t h a n k s s i r .