What is the integral of (sin^2(x/2))/(1-cos(x/2)) ?

The integral of (sin^2(x/2))/(1-cos(x/2)) is 2sin(x/2)+x+C

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integral of (sin^2(x/2))/(1-cos(x/2))

\int \frac{sin ^{2} ( \frac{x}{2} )}{1 - cos ( \frac{x}{2} )} d x

2 sin (\frac{x}{2}) + x + C

Solution steps

\int \frac{sin ^{2} ( \frac{x}{2} )}{1 - cos ( \frac{x}{2} )} d x

Rewrite using trig identities

= \int cos (\frac{x}{2}) + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos (\frac{x}{2}) d x + \int 1 d x

\int cos (\frac{x}{2}) d x = 2 sin (\frac{x}{2})

\int 1 d x = x

= 2 sin (\frac{x}{2}) + x

Add a constant to the solution

= 2 sin (\frac{x}{2}) + x + C

What is the integral of (sin^2(x/2))/(1-cos(x/2)) ?
The integral of (sin^2(x/2))/(1-cos(x/2)) is 2sin(x/2)+x+C