What is the integral of csc(ax) ?

The integral of csc(ax) is 1/a ln|tan((ax)/2)|+C

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integral of csc(ax)

\int csc (a x) d x

\frac{1}{a} ln tan (\frac{a x}{2}) + C

Solution steps

\int csc (a x) d x

Rewrite using trig identities

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

Apply u-substitution

= \int \frac{1}{a sin ( u )} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int \frac{1}{sin ( u )} d u

Apply u-substitution

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{a} ln ∣ v ∣

Substitute back

= \frac{1}{a} ln tan (\frac{a x}{2})

Add a constant to the solution

= \frac{1}{a} ln tan (\frac{a x}{2}) + C

What is the integral of csc(ax) ?
The integral of csc(ax) is 1/a ln|tan((ax)/2)|+C