Question

Find ${\int }_0^{\frac{\pi }{2}}co{s}^{3}xdx$ =

• A. $\frac{2}{3}$
• B. $2$
• C. $1$
• D. $-2$

Answer 1

The correct option is A $\frac{2}{3}$

$I={\int }_0^{\frac{\pi }{2}}co{s}^{3}xdx$

$={\int }_0^{\frac{\pi }{2}}co{s}^{2}x.\cos xdx$

$=\stackrel{\pi /2}{\underset{0}{\int }}(1-{\sin }^{2}x)\cos xdx(co{s}^{2}x+si{n}^{2}x=1)$
Tanking $\sin x=t$
$\therefore \cos xdx=dt$
If $x=0$ then $t=\sin 0$
$\therefore t=0$
And if $x=\frac{\pi }{2}$ then $t=\sin \frac{\pi }{2}$
$\therefore t=1$
$\therefore I=\stackrel{1}{\underset{0}{\int }}(1-t^2)dx$
$={(t-\frac{t^3}{3})}_0^1$
$=(1-\frac{1}{3})-(0-0)$
$\therefore I=\frac{2}{3}$.