Question

Evalute $\underset{0}{\overset{a}{\int }}x{\left(a^2-x^2\right)}^{\frac{7}{2}}dx$

Answer 1

We have,

$I={{\int }_0^{a}x(a^2-x^2)}^{\frac{7}{2}}dx$

Let

$t=a^2-x^2$

$\frac{dt}{dx}=0-2x$

$-\frac{dt}{2}=xdx$

Therefore,

$I=-\frac{1}{2}{{\int }_{a^2}^0\left(t\right)}^{\frac{7}{2}}dt$

$I=-\frac{1}{2}{\left[\frac{t^{\frac{7}{2}+1}}{\frac{7}{2}+1}\right]}_{a^2}^0$

$I=-{\left[\frac{t^{\frac{9}{2}}}{9}\right]}_{a^2}^0$

$I=-\frac{1}{9}[0^{{}^{\frac{9}{2}}}-{\left(a^2\right)}^{{}^{\frac{9}{2}}}]$

$I=\frac{a^9}{9}$

Hence, the value is $\frac{a^9}{9}$.