Using properties of definite integrals, prove that $\int_{0}^{\pi} \frac{x \tan x}{\sec x \operatorname{cosec} x} d x=\frac{\pi^{2}}{4}$

CBSE Class 12 · 2007 · Definite IntegralsLong Answer

Using properties of definite integrals, prove that

\int_{0}^{π} \frac{x t a n x}{s e c x cosec x} d x = \frac{π ^{2}}{4}

Answer: Proved

First simplify the integrand:

\frac{tan x}{sec x cosec x} = \frac{sin x / cos x}{( 1/ cos x ) ( 1/ sin x )} = sin^{2} x

. So

I = \int_{0}^{π} x sin^{2} x d x

.
Apply

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

with

a = π

and

sin^{2} (π - x) = sin^{2} x

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

Adding the two forms:

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

2 I = \frac{π ^{2}}{2}

, i.e.

I = \frac{π ^{2}}{4}

. Proved.

CBSE Class 12 Board · 2007 · Subjective