What is the integral of csc(x/2)cot(x/2) ?

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

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integral of csc(x/2)cot(x/2)

\int csc (\frac{x}{2}) cot (\frac{x}{2}) d x

- 2 csc (\frac{x}{2}) + C

Solution steps

\int csc (\frac{x}{2}) cot (\frac{x}{2}) d x

Apply u-substitution

: \int csc (u) cot (u) \cdot 2 d u

= \int csc (u) cot (u) \cdot 2 d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int csc (u) cot (u) d u

Apply u-substitution

: \int - 1 d v

= 2 \cdot \int - 1 d v

Integral of a constant:

\int a d x = a x

= 2 (- 1) v

Substitute back

= 2 (- 1) csc (\frac{x}{2})

Simplify

= - 2 csc (\frac{x}{2})

Add a constant to the solution

= - 2 csc (\frac{x}{2}) + C

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