What is the integral of csc^2(1/2 x) ?

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

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integral of csc^2(1/2 x)

\int csc^{2} (\frac{1}{2} x) d x

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

Solution steps

\int csc^{2} (\frac{1}{2} x) d x

Rewrite using trig identities

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

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

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

Apply u-substitution

= \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

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