Question

Evaluate the definite integral: $∫^\frac{π}{2}_0 cos^2 x dx$

Answer 1

Let I=$∫^\frac{π}{2}_0 cos^2 x dx$

$∫cos^2 x dx=∫(\frac{1+cos2x}{2})dx=\frac{x}{2}+\frac{sin2x}{4}=\frac{1}{2}(x+\frac{sin2x}{2})=F(x)$

By second fundamental theorem of calculus,we obtain

$I=[F(\frac{π}{2})-F(0)]$

$=\frac{1}{2}[(\frac{π}{2})-\frac{sinπ}{2})-(0+\frac{sin0}{2})]$

$=\frac{1}{2}[\frac{π}{2}+0-0-0]$

$=\frac{π}{4}$