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

Question Number 17011 by arnabpapu550@gmail.com last updated on 29/Jun/17

\int_{0}^{a} x \sqrt{\frac{a^{2} - x^{2}}{a^{2} + x^{2}}} dx

Answered by sma3l2996 last updated on 29/Jun/17

I = \int_{0}^{a} \frac{x . \sqrt{a^{2} - x^{2}}}{\sqrt{a^{2} + x^{2}}} d x

t^{2} = a^{2} - x^{2} \Rightarrow t d t = - x d x

I = - \int_{a}^{0} \frac{t}{\sqrt{2 a^{2} - t^{2}}} d t = {[\sqrt{2 a^{2} - t^{2}}]}_{a}^{0}

I = a \sqrt{2} - a \Leftrightarrow I = a (\sqrt{2} - 1)

Commented by arnabpapu550@gmail.com last updated on 30/Jun/17

The answer is \frac{a^{2}}{2} (\frac{Π}{2} - 1)

Commented by sma3l2996 last updated on 30/Jun/17

I = \int_{a}^{0} \frac{t}{\sqrt{2 a^{2} - t^{2}}} (- t d t) = \int_{0}^{a} \frac{t^{2}}{\sqrt{2 a^{2} - t^{2}}} d t

= - \int_{0}^{a} \frac{2 a^{2} - t^{2} - 2 a^{2}}{\sqrt{2 a^{2} - t^{2}}} d t = 2 a^{2} \int_{0}^{a} \frac{d t}{\sqrt{2 a^{2} - t^{2}}} - \int_{0}^{a} \sqrt{2 a^{2} - t^{2}} d t

= 2 a^{2} \int_{0}^{a} \frac{d t}{\sqrt{2} a . \sqrt{1 - {(\frac{t}{\sqrt{2} a})}^{2}}} - \sqrt{2} a \int_{0}^{a} \sqrt{1 - {(\frac{t}{\sqrt{2} a})}^{2}} d t

y = \frac{t}{\sqrt{2} a} \Rightarrow d t = \sqrt{2} a . d y

I = 2 a^{2} \int_{0}^{\frac{1}{\sqrt{2}}} \frac{d y}{\sqrt{1 - y^{2}}} - 2 a^{2} \int_{0}^{\frac{1}{\sqrt{2}}} \sqrt{1 - y^{2}} d y

I = 2 a^{2} {[{s i n}^{- 1} y]}_{0}^{\frac{\sqrt{2}}{2}} - 2 a^{2} \int_{0}^{\frac{\sqrt{2}}{2}} \sqrt{1 - y^{2}} d y

s i n (u) = \sqrt{1 - y^{2}} \Rightarrow c o s (u) d u = \frac{- y d y}{s i n (u)}

s i n (u) . \sqrt{1 - {s i n}^{2} (u)} d u = - y d y

s i n (u) . \sqrt{1 - 1 + y^{2}} d u = - y d y \Leftrightarrow s i n (u) . y . d u = - y d y

s i n (u) d u = - d y

I = 2 a^{2} \frac{π}{4} - 2 a^{2} \int_{\frac{π}{2}}^{\frac{π}{4}} s i n (u) . (- s i n (u) d u)

= a^{2} \frac{π}{2} + 2 a^{2} \int_{\frac{π}{2}}^{\frac{π}{4}} {s i n}^{2} (u) d u = a^{2} \frac{π}{2} + a^{2} \int_{\frac{π}{2}}^{\frac{π}{4}} (1 - c o s (2 u)) d u

= a^{2} \frac{π}{2} + a^{2} {[u - \frac{s i n (2 u)}{2}]}_{\frac{π}{2}}^{\frac{π}{4}} = a^{2} \frac{π}{2} + a^{2} (- \frac{π}{4} - \frac{1}{2}) = a^{2} (\frac{π}{2} - \frac{π}{4} - \frac{1}{2})

I = \frac{a^{2}}{2} (\frac{π}{2} - 1)