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Question Number 56345 by maxmathsup by imad last updated on 14/Mar/19

l e t f (a) = \int_{0}^{\infty} \frac{d x}{x^{n} + a^{n}} w i t h n i n t e g r ⩾ 2 a n d a > 0

1) c a l c u l a t e f (a) i n t e m s o f a

2) l e t g (a) = \int_{0}^{\infty} \frac{d x}{{(x^{n} + a^{n})}^{2}} c a l c u l a t e g (a) i n t e r m s o f a

3) f i n d t h e v a l u e s o f i n t e g r a l s \int_{0}^{\infty} \frac{d x}{x^{8} + 16} a n d \int_{0}^{\infty} \frac{d x}{{(x^{8} + 16)}^{2}}

Commented by maxmathsup by imad last updated on 17/Mar/19

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

\int_{0}^{\infty} \frac{d t}{t^{n} + 1} =_{t = u^{\frac{1}{n}}} \int_{0}^{\infty} \frac{1}{n (1 + u)} u^{\frac{1}{n} - 1} d u = \frac{1}{n} \int_{0}^{\infty} \frac{u^{\frac{1}{n} - 1}}{1 + u} d u

= \frac{1}{n} \frac{π}{s i n (\frac{π}{n})} = \frac{π}{n s i n (\frac{π}{n})} (w e h a v e p r o v e d t h a t \int_{0}^{\infty} \frac{x^{a - 1}}{1 + x} d x = \frac{π}{s i n (π a)} w i t h 0 < a < 1) \Rightarrow

★ f (a) = \frac{π}{n a^{n - 1} s i n (\frac{π}{n})} ★ w i t h n ⩾ 2

2) w e h a v e f^{^{'}} (a) = \int_{0}^{\infty} \frac{\partial}{\partial a} (\frac{1}{x^{n} + a^{n}}) d x = - \int_{0}^{\infty} \frac{{n a}^{n - 1}}{{(x^{n} + a^{n})}^{2}} d x

= - {n a}^{n - 1} g (a) \Rightarrow g (a) = - \frac{f (a)}{{n a}^{n - 1}} = - \frac{1}{{n a}^{n - 1}} \frac{π}{n a^{n - 1} s i n (\frac{π}{n})} \Rightarrow

★ g (a) = - \frac{π}{n^{2} a^{2 n - 2} s i n (\frac{π}{n})} ★

3) w e h a v e \int_{0}^{\infty} \frac{d x}{x^{8} + 16} = \int_{0}^{\infty} \frac{d x}{x^{8} + {(\sqrt{2})}^{8}} \Rightarrow n = 8 a n d a = \sqrt{2} \Rightarrow

\int_{0}^{\infty} \frac{d x}{x^{8} + 16} = \frac{π}{8 {(\sqrt{2})}^{7} s i n (\frac{π}{8})} a l s o w e h a v e

\int_{0}^{\infty} \frac{d x}{{(x^{8} + 16)}^{2}} = \int_{0}^{\infty} \frac{d x}{{(x^{2} + {(\sqrt{2})}^{8})}^{2}} \Rightarrow n = 8 a n d a = \sqrt{2} a t g (a) \Rightarrow

\int_{0}^{\infty} \frac{d x}{{(x^{8} + 16)}^{2}} = \frac{π}{64 {(\sqrt{2})}^{14} s i n (\frac{π}{8})} = \frac{π}{64 . 2^{7} s i n (\frac{π}{8})} w i t h

s i n (\frac{π}{8}) = \frac{\sqrt{2 - \sqrt{2}}}{2} .