Evaluate: $\int_{0}^{\pi} \frac{x \tan x}{\sec x \operatorname{cosec} x} d x$

CBSE Class 12 · 2008 · Definite IntegralsLong Answer

Evaluate:

\int_{0}^{π} \frac{x t a n x}{s e c x cosec x} d x

Answer: $\frac{π ^{2}}{4}$

Simplify the integrand:

\frac{tan x}{sec x cosec x} = sin^{2} x

, so

I = \int_{0}^{π} x sin^{2} x d x

.
Apply

x \to π - x

(with

sin^{2} (π - x) = sin^{2} x

I = \int_{0}^{π} (π - x) sin^{2} x d x

2 I = π \int_{0}^{π} sin^{2} x d x = π \int_{0}^{π} \frac{1 - cos 2 x}{2} d x = π \cdot \frac{π}{2}

Hence

I = \frac{π ^{2}}{4}

CBSE Class 12 Board · 2008 · Subjective