If $I=\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} dx$, then $\int_{0}^{\pi/2} \frac{x \sin x \cos x}{\sin ^{4} x+\cos ^{4} x} dx$ equals:

Definite Integrals — JEE Main · 2025 · 23 Jan 2025 · Shift 2 Question with Solution

I = \int_{0}^{\frac{π}{2}} \frac{s i n ^{\frac{3}{2}} x}{s i n ^{\frac{3}{2}} x + c o s ^{\frac{3}{2}} x} d x

,
then

\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

equals:

Answer: (A)

For

I = \int_{0}^{π /2} \frac{sin ^{3/2} x}{sin ^{3/2} x + cos ^{3/2} x} d x

, replace

x \to \frac{π}{2} - x

I = \int_{0}^{π /2} \frac{cos ^{3/2} x}{cos ^{3/2} x + sin ^{3/2} x} d x

Adding the two forms of

I

2 I = \int_{0}^{π /2} d x = \frac{π}{2} \Rightarrow I = \frac{π}{4}

For

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

, replace

x \to \frac{π}{2} - x

(the denominator is unchanged since

sin \leftrightarrow cos

I_{2} = \int_{0}^{π /2} \frac{( \frac{π}{2} - x ) sin x cos x}{sin ^{4} x + cos ^{4} x} d x

Adding:

2 I_{2} = \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

and substitute

t = tan^{2} x

d t = 2 tan x sec^{2} x d x

\int_{0}^{π /2} \frac{sin x cos x}{sin ^{4} x + cos ^{4} x} d x = \int_{0}^{π /2} \frac{tan x sec ^{2} x}{tan ^{4} x + 1} d x = \frac{1}{2} \int_{0}^{\infty} \frac{d t}{t ^{2} + 1} = \frac{1}{2} \cdot \frac{π}{2} = \frac{π}{4}

2 I_{2} = \frac{π}{2} \cdot \frac{π}{4} = \frac{π ^{2}}{8} \Rightarrow I_{2} = \frac{π ^{2}}{16}

JEE Main · 2025 · 23 Jan 2025 · Shift 2 · Single correct option