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

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

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

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

- x - 2 cot (x) + 2 csc (x) + C

Solution steps

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

Rewrite using trig identities

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

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

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

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

\int 1 d x = x

= - (- 2 (- cot (x) - x + csc (x))) + x

Simplify

- (- 2 (- cot (x) - x + csc (x))) + x : - x - 2 cot (x) + 2 csc (x)

= - x - 2 cot (x) + 2 csc (x)

Add a constant to the solution

= - x - 2 cot (x) + 2 csc (x) + C

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