Indefinite Integration
Trigonometric Integrals
Premium Question
Grade 12
Question:
$\int \sin^3 x \cos^3 x \, dx$ equals:
(1) $\frac{\sin^4 x}{4} - \frac{\sin^6 x}{6} + C$
(2) $\frac{\sin^4 x}{4} + \frac{\sin^6 x}{6} + C$
(3) $\frac{\sin^6 x}{6} - \frac{\sin^4 x}{4} + C$
(4) $\frac{\cos^4 x}{4} - \frac{\cos^6 x}{6} + C$
Step-by-Step Solution
Key Concept: Write $\cos^3 x = (1 - \sin^2 x) \cos x$ and substitute $u = \sin x$, turning the integral into a polynomial integral in $u$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)