Question

Evaluate : $\int sin^{3 }\,x\, cos^{3}\, x\, dx$

• A. $\frac{1}{32}\left\{\frac{3}{2} cos \,2x - \frac{1}{6}cos\, 6x\right\} +C $
• B. $\frac{1}{32}\left\{-\frac{3}{2} cos \,2x + \frac{1}{6}cos\, 6x\right\} +C $
• C. $\frac{1}{32}\left\{-\frac{3}{2} cos \,2x - \frac{1}{6}cos\, 6x\right\} +C $
• D. None of these

Answer 1

Answer: (B)

Let $I = \int sin^{3 }\,x cos^{3} \,x dx$. then,

$I= \frac{1}{8}\int\left(2 \,sin\,x cos\,x\right)^{3}dx$

$\Rightarrow I =\frac{1}{8}\int sin^{3} \,2x \,dx \Rightarrow I = \frac{1}{8}\int\frac{3sin\,2x-sin\,6x}{4}dx $

$\Rightarrow I =\frac{1}{32}\int \left(3 sin\,2x-sin \,6x\right) dx $

$= \frac{1}{32}\left\{-\frac{3}{2} cos \,2x + \frac{1}{6}cos \,6x\right\} +C $