1-a-2-x-2-3-2-solve-the-integration-

Question Number 28703 by students last updated on 29/Jan/18

\frac{1}{{(a^{2} + x^{2})}^{3 / 2}} s o l v e t h e i n t e g r a t i o n

Commented by abdo imad last updated on 29/Jan/18

l e t p u t I = \int \frac{d x}{{(a^{2} + x^{2})}^{\frac{3}{2}}} i f a \neq 0 t h e c h . x = a t a n θ g i v e

I = \int \frac{1}{(a^{2} {(1 + {t a n}^{2} θ)}^{\frac{3}{2}}} a (1 + {t a n}^{2} θ)

= \frac{1}{a^{2}} \int {(1 + {t a n}^{2} θ)}^{1 - \frac{3}{2}} d θ = \frac{1}{a^{2}} \int \frac{d θ}{\sqrt{1 + {t a n}^{2} θ}}

= \frac{ξ}{a^{2}} \int c o s θ d θ = \frac{ξ}{a^{2}} s i n θ + k = \frac{ξ}{a^{2}} s i n (a r c t a n (\frac{x}{a})) + k

a n d ξ^{2} = 1 .

i f a = 0 I = \int x^{- 3} d x = \frac{1}{- 3 + 1} x^{- 3 + 1} + k

= \frac{- 1}{2 x^{2}} + k .