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

Question Number 37360 by math khazana by abdo last updated on 12/Jun/18

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

Commented by math khazana by abdo last updated on 13/Jun/18

l e t I = \int_{0}^{+ \infty} \frac{d x}{1 + x^{3}} c h a n g e m e n t x^{3} = t g i v e

I = \int_{0}^{\infty} \frac{1}{1 + t} \frac{1}{3} t^{\frac{1}{3} - 1} d t

= \frac{1}{3} \int_{0}^{\infty} \frac{t^{\frac{1}{3} - 1}}{1 + t} d t = \frac{1}{3} \frac{π}{s i n (\frac{π}{3})}

= \frac{1}{3} \frac{π}{\frac{\sqrt{3}}{2}} \Rightarrow I = \frac{2 π}{3 \sqrt{3}} .

I h a v e u s e d t b e f o r m u l a f o r 0 < a < 1

\int_{0}^{\infty} \frac{t^{a - 1}}{1 + t} d t = \frac{π}{s i n (π a)} .