we-give-0-t-a-1-1-t-1-dt-pi-sin-pia-1-with-0-lt-a-lt-1-find-the-value-of-0-1-x-16-1-dx-

Question Number 25682 by abdo imad last updated on 13/Dec/17

w e g i v e \int_{0}^{\infty} t^{a - 1} {(1 + t)}^{- 1} d t = π {(s i n (π a))}^{- 1} w i t h 0 < a < 1 f i n d t h e v a l u e o f \int_{0}^{\infty} {(1 + x^{16})}^{- 1} d x

Answered by ajfour last updated on 13/Dec/17

I f x^{16} = t a n d l e t \int \frac{d x}{1 + x^{16}} = I

T h e n I = \int \frac{d t}{16 x^{15} (1 + t)}

= \frac{1}{16} \int_{0}^{\infty} \frac{t^{- 15 / 16} d t}{1 + t}

= \frac{1}{16} \int_{0}^{\infty} \frac{t^{\frac{1}{16} - 1} d t}{1 + t} = \frac{1}{16} \times \frac{π}{\sin (\frac{π}{16})}

\sin \frac{π}{16} = \sqrt{\frac{1 - \cos (π / 8)}{2}}

= \sqrt{\frac{1 - \sqrt{\frac{1 + 1 / \sqrt{2}}{2}}}{2}} .

Facebook Tweet Pin