Question

The value of $\underset{0}{\overset{\pi /2}{\int }}{\sin }^{10}x\cdot {\cos }^{2}xdx$ is

• A. $\frac{13\pi }{559}$
• B. $\frac{21\pi }{1024}$
• C. $\frac{13\pi }{1098}$
• D. $\frac{21\pi }{2048}$

Answer 1

The correct option is D $\frac{21\pi }{2048}$

Applying walli's formula in $I_{10,2}=\underset{0}{\overset{\pi /2}{\int }}{\sin }^{10}x\cdot {\cos }^{2}xdx{I}_{10,2}=\frac{9\cdot 7\cdot 5\cdot 3}{12\cdot 10\cdot 8\cdot 6\cdot 4\cdot 2}\cdot \frac{\pi }{2}\{\because I_{m,n}=\frac{(m-1)(m-3)(m-5)\cdots (n-1)(n-3)(n-5)}{(m+n)(m+n-2)(m+n-4)\cdots }\cdot \frac{\pi }{2}\text{, when both}m,n\text{is even}\}\Rightarrow I_{10,2}=\frac{21\pi }{2048}$