Evaluate the following integrals:
Question:

Evaluate the following integrals:

$\int \frac{x^{7}}{\left(a^{2}-x^{2}\right)^{5}} d x$

Solution:

PUT $x=a \sin \theta$, so $d x=a \cos \theta d \theta$ and $\theta=\sin ^{-}(x / a)$

Above equation becomes,

$\int \frac{\mathrm{x}^{7}}{\left(\mathrm{a}^{2}-\mathrm{x}^{2}\right)^{5}} \mathrm{dx}==\int \frac{\mathrm{a}^{7} \sin ^{\prime} \theta}{\left(\mathrm{a}^{2}-\mathrm{a}^{2} \sin ^{2} \theta\right)^{5}}(\operatorname{acos} \theta \mathrm{d} \theta)=\int \frac{\mathrm{a}^{7} \sin ^{\prime} \theta}{\left(\mathrm{a}^{2}\right)^{5}\left(1-\sin ^{2} \theta\right)^{5}}(\operatorname{acos} \theta \mathrm{d} \theta)\left\{\right.$ take $\mathrm{a}^{2}$ outside $)$

$=\int \frac{\mathrm{a}^{7} \sin ^{\prime} \theta}{\left(\mathrm{a}^{2}\right)^{5}\left(1-\sin ^{2} \theta\right)^{5}}(\mathrm{a} \cos \theta \mathrm{d} \theta)=\int \frac{\mathrm{a}^{7} \sin ^{7} \theta}{\left(\mathrm{a}^{2 \mathrm{o}}\left(1-\sin ^{2} \theta\right)^{5}\right.}(\mathrm{acos} \theta \mathrm{d} \theta)$

$=\frac{1}{\mathrm{a}^{2}} \int \frac{1}{\cos ^{2} \theta} \mathrm{d} \theta=\frac{1}{\mathrm{a}^{2}} \int \sec ^{2} \theta \mathrm{d} \theta=\frac{1}{\mathrm{a}^{2}}(\tan \theta+\mathrm{c})$

Put $\theta=\sin ^{-}(x / a)$

$=\frac{1}{a^{2}}\left(\tan \sin ^{-}\left(\frac{x}{a}\right)+c\right)$