Evaluate: $\int_{0}^{\pi / 2} \frac{x \sin x \cos x}{\sin ^{4} x+\cos ^{4} x} d x$

CBSE Class 12 · 2011 · Definite IntegralsLong Answer

Evaluate:

\int_{0}^{π /2} \frac{x s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x

Answer: $\frac{π ^{2}}{16}$

Let

I = \int_{0}^{π /2} \frac{x sin x cos x}{sin ^{4} x + cos ^{4} x} d x

. Apply

x \to \frac{π}{2} - x

(which swaps

sin \leftrightarrow cos

, leaving the denominator unchanged) and add:

2 I = \frac{π}{2} \int_{0}^{π /2} \frac{sin x cos x}{sin ^{4} x + cos ^{4} x} d x

Divide numerator and denominator by

cos^{4} x

to get

\frac{tan x sec ^{2} x}{1 + tan ^{4} x}

. Put

z = tan^{2} x \Rightarrow d z = 2 tan x sec^{2} x d x

;

x = 0 \to z = 0

x = \frac{π}{2} \to z = \infty

2 I = \frac{π}{2} \cdot \frac{1}{2} \int_{0}^{\infty} \frac{d z}{1 + z ^{2}} = \frac{π}{4} [tan^{- 1} z]_{0}^{\infty} = \frac{π}{4} \cdot \frac{π}{2} = \frac{π ^{2}}{8}

Hence

I = \frac{π ^{2}}{16}

CBSE Class 12 Board · 2011 · Subjective