First slide

Methods of integration

Question

$\int \frac{\sin^{2} x \cos^{2} x d x}{{(\sin^{5} x + \cos^{3} x \sin^{2} x + \sin^{3} \cos^{2} x + \cos^{5} x)}^{2}}$

Difficult

A

$\frac{- 1}{1 + \cos^{3} x} + C$
B

$\frac{1}{3 (1 + \tan^{3} x)} + C$
C

$\frac{- 1}{3 (1 + \tan^{3} x)} + C$
D

$\frac{1}{1 + \cot^{3} x} + C$

Solution

$\begin{array}{l} \int \frac{\sin^{2} x \cos^{2} x d x}{{[(\sin^{3} x + \cos^{3} x) (\sin^{2} x + \cos^{2} x)]}^{2}} = \int \frac{\sin^{2} x \cos^{2} x d x}{{(\sin^{3} x + \cos^{3} x)}^{2}} = \int \frac{\tan^{2} x \sec^{2} x d x}{{(1 + \tan^{3} x)}^{2}} put t = 1 + \tan^{3} x \\ = \frac{1}{3} \int \frac{d t}{t^{2}} = \frac{- 1}{3 t} + C \end{array}$

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