Question

Evaluate the integral
${\int }_0^{2\pi }{\cos }^{6}xdx$

• A. $\frac{5\pi }{16}$
• B. $\frac{35\pi }{128}$
• C. $\frac{5\pi }{8}$
• D. $\frac{5\pi }{18}$

Answer 1

The correct option is C $\frac{5\pi }{8}$

${\int }_0^{2\pi }{\cos }^{6}xdx=2{\int }_0^{\pi }{\cos }^{6}xdx$

$=2\times 2{\int }_0^{\pi /2}{\cos }^{6}xdx(\because {\int }_0^{2\pi }f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx\text{if}f(2a–x)=fx\left)\right)$

$=4\times \frac{5\times 3\times 1}{6\times 4\times 2}\times \frac{\pi }{2}$ (Waali’s formula)

$=\frac{5\pi }{8}$.