Definite Integration
Integration by Substitution
Premium Question
Grade 12
Question:
$\int_0^1 x^2 \sqrt{1 - x^2} \, dx$ equals:
(1) $\frac{\pi}{16}$
(2) $\frac{\pi}{8}$
(3) $\frac{\pi}{4}$
(4) $\frac{1}{4}$
Step-by-Step Solution
Key Concept: Substitute $x = \sin \theta$ so that $\sqrt{1 - x^2} = \cos \theta$, and rewrite the resulting product $\sin^2 \theta \cos^2 \theta$ using the double-angle identity $\sin \theta \cos \theta = \frac{1}{2} \sin 2\theta$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)