$\int_{0}^{\pi/4} \frac{\cos^2 x \sin^2 x}{(\cos^3 x+\sin^3 x)^2}\,dx$ is equal to

JEE Main · 2024 · 06 Apr 2024 · Shift 1 · Definite Integrals

\int_{0}^{π /4} \frac{c o s ^{2} x s i n ^{2} x}{( c o s ^{3} x + s i n ^{3} x ) ^{2}} d x

is equal to

Answer: (C)

Divide numerator and denominator by

cos^{6} x

\int_{0}^{π /4} \frac{tan ^{2} x sec ^{2} x}{( 1 + tan ^{3} x ) ^{2}} d x

.
Let

t = 1 + tan^{3} x

d t = 3 tan^{2} x sec^{2} x d x

; limits

1 \to 2

= \frac{1}{3} \int_{1}^{2} \frac{d t}{t ^{2}} = \frac{1}{3} [- \frac{1}{t}]_{1}^{2} = \frac{1}{3} (1 - \frac{1}{2}) = \frac{1}{6} .

JEE Main · 2024 · 06 Apr 2024 · Shift 1 · Single correct option

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