What is the integral of csc^2(-r/2) ?

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

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integral of csc^2(-r/2)

\int csc^{2} (- \frac{r}{2}) d r

- 2 cot (\frac{r}{2}) + C

Solution steps

\int csc^{2} (- \frac{r}{2}) d r

Simplify

csc^{2} (- \frac{r}{2}) : csc^{2} (\frac{r}{2})

= \int csc^{2} (\frac{r}{2}) d r

Apply u-substitution

: \int csc^{2} (u) \cdot 2 d u

= \int csc^{2} (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^{2} (u) d u

Use the common integral:

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

= 2 (- cot (u))

Substitute back

u = \frac{r}{2}

= 2 (- cot (\frac{r}{2}))

Simplify

= - 2 cot (\frac{r}{2})

Add a constant to the solution

= - 2 cot (\frac{r}{2}) + C

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