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

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

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

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

- \frac{1}{2} x - \frac{1}{2} sin (x) - cot (x) + C

Solution steps

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

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

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

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

Simplify

\frac{1}{2} (x - sin (x)) - x - cot (x) : - \frac{1}{2} x - \frac{1}{2} sin (x) - cot (x)

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

Add a constant to the solution

= - \frac{1}{2} x - \frac{1}{2} sin (x) - cot (x) + C

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