calculate-1-x-3-1-x-6-dx-

Question Number 67528 by mathmax by abdo last updated on 28/Aug/19

c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{1 + x^{3}}{1 + x^{6}} d x

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

l e t n a m e d i t K

w h e n c h a n g i n g x = - u

K = \int_{- \infty}^{\infty} \frac{1 - u^{3}}{1 + u^{6}} d u

S o 2 K = \int_{- \infty}^{\infty} \frac{2 d u}{1 + u^{6}} = 4 \int_{0}^{\infty} \frac{d u}{1 + u^{6}}

c h a n g e u = v^{\frac{1}{6}} \Rightarrow d u = \frac{v^{\frac{1}{6} - 1}}{6} d v

K = \frac{2}{6} \int_{0}^{\infty} \frac{v^{\frac{1}{6} - 1}}{1 + v} d v = \frac{π}{3 s i n (\frac{π}{6})} = \frac{2 π}{3}

Commented by mathmax by abdo last updated on 29/Aug/19

t h a n k y o u s i r .

Commented by mathmax by abdo last updated on 29/Aug/19

I = \int_{- \infty}^{+ \infty} \frac{1 + x^{3}}{1 + x^{6}} d x \Rightarrow I = \int_{- \infty}^{+ \infty} \frac{d x}{1 + x^{6}} + \int_{- \infty}^{+ \infty} \frac{x^{3}}{1 + x^{6}} d x

b u t \int_{- \infty}^{+ \infty} \frac{x^{3}}{1 + x^{6}} d x = 0 (t h e f u n c t i o n i s o d d) \Rightarrow I = 2 \int_{0}^{\infty} \frac{d x}{1 + x^{6}}

c h a 7 g e m e n t x = t^{\frac{1}{6}} g i v e I = 2 \int_{0}^{\infty} \frac{1}{1 + t} \times \frac{1}{6} t^{\frac{1}{6} - 1} d t

= \frac{1}{3} \int_{0}^{\infty} \frac{t^{\frac{1}{6} - 1}}{1 + t} d t = \frac{1}{3} \frac{π}{s i n (\frac{π}{6})} = \frac{π}{3 \times \frac{1}{2}} = \frac{2 π}{3} .