find-0-x-5-1-x-7-dx-

Question Number 42810 by maxmathsup by imad last updated on 02/Sep/18

f i n d \int_{0}^{\infty} \frac{x^{5}}{1 + x^{7}} d x .

Commented by prof Abdo imad last updated on 03/Sep/18

c h a n g e m e n t x = t^{\frac{1}{7}} g i v e

I = \int_{0}^{\infty} \frac{t^{\frac{5}{7}}}{1 + t} \frac{1}{7} t^{\frac{1}{7} - 1} d t

= \frac{1}{7} \int_{0}^{\infty} \frac{t^{\frac{6}{7} - 1}}{1 + t} d t = \frac{1}{7} \frac{π}{s i n (\frac{6 π}{7})} = \frac{π}{7 s i n (\frac{π}{7})} .

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Sep/18

x^{7} = {t a n}^{2} α x = {(t a n α)}^{\frac{2}{7}}

d x = \frac{2}{7} {t a n}^{\frac{- 5}{7}} α {s e c}^{2} α d α

\int_{0}^{\frac{Π}{2}} \frac{{(t a n α)}^{\frac{10}{7}}}{{s e c}^{2} α} \times \frac{2}{7} {(t a n α)}^{\frac{- 5}{7}} \times {s e c}^{2} α d α

\frac{2}{7} \int_{0}^{\frac{Π}{2}} {(t a n α)}^{\frac{5}{7}} α d α

\frac{2}{7} \int_{0}^{\frac{Π}{2}} {(s i n α)}^{\frac{5}{7}} {(c o s α)}^{\frac{- 5}{7}} d α

u s i n g f o r m u l a

\int_{0}^{\frac{Π}{2}} {s i n}^{2 p - 1} α {c o s}^{2 q - 1} α d α = \frac{⌈ p \times ⌈ q}{2 ⌈ (p + q)}

h e r e 2 p - 1 = \frac{5}{7} 2 p = \frac{12}{7} p = \frac{6}{7}

2 q - 1 = - \frac{5}{7} 2 q = \frac{2}{7} q = \frac{1}{7}

= \frac{2}{7} \int_{0}^{\frac{Π}{2}} {(s i n α^{})}^{2 \times \frac{6}{7} - 1} {(c o s α)}^{\frac{2 \times 1}{7} - 1} d α

\frac{2}{7} \times \frac{⌈ (\frac{6}{7}) \times ⌈ (\frac{1}{7})}{2 ⌈ (\frac{6}{7} + \frac{1}{7})} = \frac{1}{7} \times ⌈ (\frac{1}{7}) \times ⌈ (1 - \frac{1}{7}) = \frac{1}{7} \times \frac{Π}{s i n (\frac{Π}{7})}

⌈ (p) \times ⌈ (1 - p) = \frac{Π}{s i n (p Π)}