dx is equal to

Engineering

Mathematics

Introduction to Determinants

Question

$\int_{0}^{π / 4} \frac{\cos^{2} {xsin}^{2} x}{{(\cos^{3} x + \sin^{3} x)}^{2}}$ dx is equal to

$\frac{1}{6}$

$\frac{1}{12}$

$\frac{1}{6}$

$\frac{1}{3}$

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Divide by cos⁶x in Nr and Dr.

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

1 + tan³x = t

3tan²xsec²x dx = dt

$= \frac{1}{3} \int_{1}^{2} \frac{dt}{t^{2}}$

$= \frac{1}{3} {(\frac{- 1}{t})}_{1}^{2} = \frac{1}{3} [\frac{- 1}{2} + 1] = \frac{1}{6}$

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