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

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

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integral of csc^2(x/2)

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

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

Solution steps

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

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{x}{2}

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

Simplify

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

Add a constant to the solution

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

\frac{s + 5}{s ^{2} + 4 s + 3}

\frac{d}{d x} (e^{x} + c)

f (x) = x^{2} + x + 1

y^{^{'''}} - 6 y^{^{''}} + 8 y^{^{'}} = 0

\int \frac{1}{p - p ^{2}} d p