What is the integral of $int sin^2(πx / 2)$ ?

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Answer 1 · 2017-03-14T03:00:13

$1/2x-1/(2pi)sin(pix)+C$

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 $cos(pix)$ with substitution:

$=1/2intdx-1/2intcos(pix)dx$

Let $u=pix$ to $du=pidx$ :

$=1/2x-1/(2pi)intcos(pix)(pidx)=1/2x-1/(2pi)intcos(u)du$

$=1/2x-1/(2pi)sin(pix)+C$

What is the integral of int sin^2(πx / 2)int sin^2(πx / 2)?