x-2-a-bx-2-5-dx-where-a-b-gt-0-

Question Number 128334 by liberty last updated on 06/Jan/21

Ω = \int \frac{x^{2}}{\sqrt{{(a + {bx}^{2})}^{5}}} dx; where : a; b > 0

Answered by bramlexs22 last updated on 06/Jan/21

Ω = \int \frac{x^{2}}{{(a + {b x}^{2})}^{5 / 2}} d x

Ω = \int \frac{x^{2}}{{(x^{2} ({a x}^{- 2} + b))}^{5 / 2}} d x

Ω = \int \frac{x^{- 3}}{{({a x}^{- 2} + b)}^{5 / 2}} d x

s e t t i n g {a x}^{- 2} + b = v

w e g e t - 2 {a x}^{- 3} d x = d v

t h e n Ω = - \frac{1}{2 a} \int \frac{d v}{v^{5 / 2}}

Ω = - \frac{1}{2 a} \int v^{- 5 / 2} d v = - \frac{1}{2 a} . (- \frac{2}{3}) v^{- 3 / 2} + C

Ω = \frac{1}{3 a} . \frac{1}{\sqrt{v^{3}}} + C = \frac{1}{3 a \sqrt{{(\frac{a + {b x}^{2}}{x^{2}})}^{3}}} + C

Ω = \frac{x^{3}}{3 a \sqrt{{(a + {b x}^{2})}^{3}}} + C