Question

$\underset{-\pi /2}{\overset{\pi /2}{\int }}{\sin }^{4}x{\cos }^{6}xdx$ is equal to

• A. $\frac{3\pi }{64}$
• B. $\frac{3\pi }{572}$
• C. $\frac{3\pi }{256}$
• D. $\frac{3\pi }{128}$

Answer 1

The correct option is C $\frac{3\pi }{256}$

$I=\underset{-\pi /2}{\overset{\pi /2}{\int }}{\sin }^{4}x{\cos }^{6}xdx=2\underset{0}{\overset{\pi /2}{\int }}{\sin }^{4}x{\cos }^{6}xdx$
$\{\because {\sin }^{4}x{\cos }^{6}x\text{is an even function}\}$
We know according to Wallis formula, when both $m$ and $n$ are even,
$I_{m,n}=\frac{(m-1)(m-3)(m-5)\cdots (n-1)(n-3)(n-5)}{(m+n)(m+n-2)(m+n-4)\cdots }\cdot \frac{\pi }{2}\therefore I=2I_{4,6}=2\frac{(3\cdot 1)(5\cdot 3\cdot 1)}{10\cdot 8\cdot 6\cdot 4\cdot 2}\cdot \frac{\pi }{2}=\frac{3\pi }{256}$