What is the integral of #int sin^2(πx / 2)#?
1 Answer
Mar 14, 2017
Explanation:
Use the form of the cosine double-angle identity with sine in it:
#cos(2alpha)=1-2sin^2(alpha)#
#sin^2(alpha)=1/2(1-cos(2alpha))#
Which implies that:
#sin^2((pix)/2)=1/2(1-cos(pix))#
Then:
#intsin^2((pix)/2)dx=1/2int(1-cos(pix))dx#
Integrating both of these, and integrating
#=1/2intdx-1/2intcos(pix)dx#
Let
#=1/2x-1/(2pi)intcos(pix)(pidx)=1/2x-1/(2pi)intcos(u)du#
#=1/2x-1/(2pi)sin(pix)+C#