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

The integral of csc((x-pi)/2)cot((x-pi)/2) is -2csc(1/2 (x-pi))+C

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

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

- 2 csc (\frac{1}{2} (x - π)) + C

Solution steps

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

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

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

Apply u-substitution

= \int csc (\frac{u}{2}) cot (\frac{u}{2}) d u

Apply u-substitution

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

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int - 1 d w

Integral of a constant:

\int a d x = a x

= 2 (- 1) w

Substitute back

= 2 (- 1) csc (\frac{x - π}{2})

Simplify

2 (- 1) csc (\frac{x - π}{2}) : - 2 csc (\frac{1}{2} (x - π))

= - 2 csc (\frac{1}{2} (x - π))

Add a constant to the solution

= - 2 csc (\frac{1}{2} (x - π)) + C

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