Question

By using the properties of definite integrals, evaluate the integral: $∫^{2π}_0 cos^5 xdx$

Answer 1

Let I=$∫^{2π}_0 cos^5 xdx...(1)$

$cos^5(2π-x)=cos^5x$

It is known that,

$∫^{2a}_0ƒ(x)dx=2∫^a_0ƒ(x)dx,if\, ƒ(2a-x)=ƒ(x)$

$=0\,\, if\,\, ƒ(2a-x)=-ƒ(x)$

$∴I=2∫^π_0cos^5 xdx$

$⇒I=2(0)=0 \,\,\,\,[cos^5(π-x)=-cos^5x]$