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

Question Number 66468 by mathmax by abdo last updated on 15/Aug/19

c a l c u l a t e I_{n} = \int_{0}^{\infty} \frac{d x}{{(x^{n} + 3)}^{2}} w i t h n > 1

Commented by mathmax by abdo last updated on 16/Aug/19

l e t f (a) = \int_{0}^{\infty} \frac{d x}{a + x^{n}} w i t h a > 0 \Rightarrow f (a) = \frac{1}{a} \int_{0}^{\infty} \frac{d x}{(1 + \frac{x^{n}}{a})}

l e t u s e t h e c h a n g e m e n t \frac{x^{n}}{a} = u^{n} \Rightarrow x^{n} = a u^{n} \Rightarrow x = a^{\frac{1}{n}} u \Rightarrow

f (a) = \frac{1}{a} \int_{0}^{\infty} \frac{1}{1 + u^{n}} a^{\frac{1}{n}} d u = a^{\frac{1}{n} - 1} \int_{0}^{\infty} \frac{d u}{1 + u^{n}} c h a n g e m e n t u = α^{\frac{1}{n}}

g i v e \int_{0}^{\infty} \frac{1}{1 + α} \frac{1}{n} α^{\frac{1}{n} - 1} d α = \frac{1}{n} \int_{0}^{\infty} \frac{α^{\frac{1}{n} - 1}}{1 + α} d α = \frac{1}{n} \frac{π}{s i n (\frac{π}{n})} \Rightarrow

f (a) = a^{\frac{1}{n} - 1} \times \frac{π}{n s i n (\frac{π}{n})} = \frac{π a^{\frac{1}{n} - 1}}{n s i n (\frac{π}{n})} a n d w e h a v e

f^{^{'}} (a) = - \int_{0}^{\infty} \frac{d x}{{(a + x^{n})}^{2}} \Rightarrow \int_{0}^{\infty} \frac{d x}{{(a + x^{n})}^{2}} = - f^{^{'}} (a)

f^{^{'}} (a) = \frac{π (\frac{1}{n} - 1) a^{\frac{1}{n} - 2}}{n s i n (\frac{π}{n})} \Rightarrow \int_{0}^{\infty} \frac{d x}{{(a + x^{n})}^{2}} = \frac{π (1 - \frac{1}{n}) a^{\frac{1}{n} - 2}}{n s i n (\frac{π}{n})}

a = 3 \Rightarrow \int_{0}^{\infty} \frac{d x}{{(3 + x^{n})}^{2}} = \frac{π (1 - \frac{1}{n}) 3^{\frac{1}{n} - 2}}{n s i n (\frac{π}{n})} = I_{n}