0-a-2-x-2-a-2-x-2-3-2-dx-

Question Number 13960 by tawa tawa last updated on 25/May/17

\int_{0}^{\frac{a}{2}} x^{2} {(a^{2} - x^{2})}^{- 3 / 2} dx

Answered by ajfour last updated on 25/May/17

I = \int_{0}^{a / 2} \frac{x^{2}}{{(a^{2} - x^{2})}^{3 / 2}} d x

l e t x = a \sin θ \Rightarrow d x = a \cos θ d θ

w h e n x = 0, θ = 0

w h e n x = \frac{a}{2}, θ = \frac{π}{6}

I = \int_{0}^{π / 6} \frac{a^{2} \sin^{2} θ}{a^{3} \cos^{3} θ} (a \cos θ d θ)

= \int_{0}^{π / 6} \tan^{2} θ d θ

= \int_{0}^{π / 6} (\sec^{2} θ - 1) d θ

= [\tan θ - θ] ∣_{0}^{π / 6}

I = \frac{1}{\sqrt{3}} - \frac{π}{6} = \frac{(2 \sqrt{3} - π)}{6} .

Commented by tawa tawa last updated on 25/May/17

God bless you sir . i understand now .