First slide

Evaluation of definite integrals

Question

$\int_{0}^{\frac{π}{2}} \frac{x \sin x \cos x}{\cos^{4} x + \sin^{4} x} =$

Difficult

A

$\frac{π^{2}}{4}$
B

$\frac{π^{2}}{8}$
C

$\frac{π^{2}}{16}$
D

$\frac{π^{2}}{2}$

Solution

$I = \int_{0}^{\frac{π}{2}} \frac{x \sin x \cos x}{\cos^{4} x + \sin^{4} x} d x, x \to \frac{π}{2} - x$
$I = \int_{0}^{\frac{π}{2}} \frac{(\frac{π}{2} - x) \cos x \sin x}{\cos^{4} x + \sin^{4} x} d x$

$2 I = \frac{π}{2} \int_{0}^{\frac{π}{2}} \frac{\cos x \sin x}{\cos^{4} x + \sin^{4} x} d x, \tan^{2} x = t$

$= \frac{π}{4} \int_{0}^{\infty} \frac{d t}{1 + t^{2}} = \frac{π}{4} \cdot \frac{π}{2} = \frac{π^{2}}{8} \to I = \frac{π^{2}}{16}$

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