calculate-0-x-3-1-x-5-dx-

Question Number 33027 by prof Abdo imad last updated on 09/Apr/18

c a l c u l a t e \int_{0}^{\infty} \frac{x^{3}}{1 + x^{5}} d x .

Answered by Joel578 last updated on 14/Apr/18

I = \int_{0}^{\infty} \frac{x^{3}}{1 + x^{5}} d x

u = x^{5} \to d u = 5 x^{4} d x

I = \frac{1}{5} \int_{0}^{\infty} \frac{x^{- 1}}{1 + u} d u = \frac{1}{5} \int_{0}^{\infty} \frac{u^{- \frac{1}{5}}}{1 + u} d u

= \frac{1}{5} \int_{0}^{\infty} \frac{u^{\frac{4}{5} - 1}}{1 + u} d u (\int_{0}^{\infty} \frac{t^{a - 1}}{1 + t} d t = \frac{π}{\sin (π a)})

= \frac{π}{5 \sin (\frac{4 π}{5})}

= \frac{4 π}{5 \sqrt{10 - 2 \sqrt{5}}} \approx 1.07

Commented by Joel578 last updated on 10/Apr/18

\sin (\frac{4 π}{5}) = \sin (\frac{π}{5}) = \frac{1}{4} \sqrt{10 - 2 \sqrt{5}}

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