Question

Evaluate : $\int si{n}^{3}xco{s}^{3}xdx$

• A. $\frac{1}{32}\left\{\frac{3}{2}cos2x-\frac{1}{6}cos6x\right\}+C$
• B. $\frac{1}{32}\left\{-\frac{3}{2}cos2x+\frac{1}{6}cos6x\right\}+C$
• C. $\frac{1}{32}\left\{-\frac{3}{2}cos2x-\frac{1}{6}cos6x\right\}+C$
• D. None of these

Answer 1

Answer: (B)

Let $I=\int si{n}^{3}xco{s}^{3}xdx$. then,

$I=\frac{1}{8}\int {\left(2sinxcosx\right)}^{3}dx$

$\Rightarrow I=\frac{1}{8}\int si{n}^{3}2xdx\Rightarrow I=\frac{1}{8}\int \frac{3sin2x-sin6x}{4}dx$

$\Rightarrow I=\frac{1}{32}\int \left(3sin2x-sin6x\right)dx$

$=\frac{1}{32}\left\{-\frac{3}{2}cos2x+\frac{1}{6}cos6x\right\}+C$