\(\int \frac{\sqrt{(9 - x^2)^3}}{x^4}dx = \int \frac{(9 - x^2)\sqrt{9 - x^2}}{x^4}dx\)
\(= \int \frac{(9 - 9\sin^2\theta)3\cos \theta .3\cos \theta d\theta}{(3\sin \theta)^4}\) Let \(x= 3\sin\theta, dx= 3\cos\theta d\theta\)
\(= \frac{81}{81}\int \frac{\cos^4\theta}{\sin^4\theta} d\theta\)
\(= \int \cot^4 \theta d\theta\)
\(= \int (cosec^2\theta - 1)\cot^2\theta d\theta\)
\(= \int \cot^2\theta \,cosec^2\theta d\theta - \int (cosec^2\theta - 1)d\theta\)
\(= - \frac{\cot^2\theta}3 + \cot \theta + \theta + C\)
\(= -\frac 13 \frac{(1 - \sin^2\theta)^{3/2}}{\sin^3 \theta} + \frac{(1 - \sin^2\theta)^{1/2}}{\sin\theta} + \sin^{-1} \left(\frac x3\right) + C\)
\(= -\frac 13 \frac{\left(1 - \left(\cfrac x3\right)^2\right)^{3/2}}{\left(\frac x3\right)^3} + \cfrac{\left(1 - \left(\frac x3\right)^2\right)^{1/2}}{\frac x3} + \sin^{-1} (\frac x3) + C\)
\(=-\frac 13 \frac{(9 - x^2 )^{3/2}}{x^3} + \frac{\sqrt{9 - x^2}}x + \sin^{-1} (\frac x3) + C\)
