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

The integral of csc(x/2) is 2ln|tan(x/4)|+C

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integral of csc(x/2)

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

2 ln tan (\frac{x}{4}) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

= \int 2 csc (u) 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) d u

Apply u-substitution

= 2 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 ln ∣ v ∣

Substitute back

= 2 ln tan (\frac{\frac{x}{2}}{2})

\frac{\frac{x}{2}}{2} = \frac{x}{4}

= 2 ln tan (\frac{x}{4})

Add a constant to the solution

= 2 ln tan (\frac{x}{4}) + C