0 / 4 cos 2 x sin 2 x ( cos 3 x + sin 3 x ) 2 ? d x is equal to [2024]

Question

$\int_{0}^{π / 4} \frac{\cos^{2} x \sin^{2} x}{{(\cos^{3} x + \sin^{3} x)}^{2}} d x$ is equal to [2024]

Smart Achievers · Accepted Answer

(1)

Let $I = \int_{0}^{π / 4} \frac{\cos^{2} x \sin^{2} x}{{(\cos^{3} x + \sin^{3} x)}^{2}} d x$

On dividing Nr. and Dr. by $\cos^{6} x,$ we get

$I = \int_{0}^{π / 4} \frac{\tan^{2} x \sec^{2} x}{{(1 + \tan^{3} x)}^{2}} d x$

Put $1 + \tan^{3} x = t$

$\Rightarrow 3 \tan^{2} x \sec^{2} x d x = d t$

when $x = 0,$ $t = 1$ and when $x = π / 4,$ $t = 2$

$∴$ $I = \frac{1}{3} \int_{1}^{2} \frac{d t}{t^{2}} = \frac{1}{3} {[\frac{- 1}{t}]}_{1}^{2} = \frac{1}{3} [\frac{- 1}{2} + 1] = \frac{1}{6}$

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