∫_0^π2 x cos x sin x dx ∫_0^π2 (π2 x) cos x sin x dx Adding these two equations, we get ∫_0^π2π4sin2x dx #160; π8 [ cos 2x]^π2_0 2I = ...

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Integration of Piecewise Continuous Functions

∫π/20x x cose...

Question

$\int_{0}^{π / 2} \frac{x}{sec x c o s e c x} d x = \frac{π}{8}$ .

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Solution

$\int_{0}^{\frac{π}{2}} x cos x sin x d x$
$\int_{0}^{\frac{π}{2}} (\frac{π}{2} - x) cos x sin x d x$
Adding these two equations, we get
$\int_{0}^{\frac{π}{2}} \frac{π}{4} sin 2 x d x$
$\frac{π}{8} [- cos 2 x]_{0}^{\frac{π}{2}}$
$2 I = \frac{π}{4}$
$I = \frac{π}{8}$

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