Definite Integration
Trigonometric Substitution
nta_pyq_2024_apr
Grade 12
Question:
$\int_0^{\pi/4}\dfrac{\cos^2x\sin^2x}{(\cos^3x+\sin^3x)^2}\,dx$ is equal to:
Step-by-Step Solution
Key Concept: Divide numerator and denominator by $\cos^6x$. Let $t=1+\tan^3x$, $dt=3\tan^2x\sec^2x\,dx$. Integral becomes $\frac{1}{3}\int_1^2\frac{dt}{t^2}$.
$\frac{1}{3}\int_1^2 t^{-2}dt=\frac{1}{3}\cdot\frac{1}{2}=\frac{1}{6}$.
Correct Answer: 1