Question

Evaluate: ${\int }_0^{\frac{\pi }{2}}{\cos }^{5}xdx$

• A. $\frac{8}{15}$
• B. $\frac{7}{15}$
• C. $\frac{1}{15}$
• D. $0$

Answer 1

The correct option is A $\frac{8}{15}$

Given ${\int }_0^{\frac{\pi }{2}}{\cos }^5\left(x\right)dx$

$={\int }_0^{\frac{\pi }{2}}{\cos }^4\left(x\right)\cos \left(x\right)dx$

$={\int }_0^{\frac{\pi }{2}}[1-{\sin }^2(x\left){]}^2\cos \right(x)dx$ .....1 $\because {\cos }^2\left(x\right)=1-{\sin }^2$(x)

Let $\sin \left(x\right)=t$ ......2

On differentiating equation 2.

$\cos \left(x\right)dx=dt$ ......3

When $x=0,t=0$; $x=\frac{\pi }{2},t=1$

Substituting values in equation 1.

${\int }_0^1[1-t^2{]}^{2}dt$

$={\int }_0^1[1+t^4-2t^2]dt$

$={\int }_0^{1}dt+{\int }_0^1{t}^{4}dt-{\int }_0^{1}2t^{2}dt$

$=[t{]}_0^1+[\frac{t^5}{5}{]}_0^1-2[\frac{t^3}{3}{]}_0^1$

$=1+\frac{1}{5}-2\left[\frac{1}{3}\right]$

$=1+\frac{1}{5}-\frac{2}{3}$

$=\frac{15+3-10}{15}$

$=\frac{8}{15}$