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

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

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integral of (sin^2(x/2))/(sin^2(x))

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{( 1 - cos ( x ) ) ^{2}}{sin ^{2} ( x )} : \frac{1}{sin ^{2} ( x )} - \frac{2 cos ( x )}{sin ^{2} ( x )} + \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

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

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

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

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

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

Simplify

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