What is the integral of 1/(sin(ax)cos(ax)) ?

The integral of 1/(sin(ax)cos(ax)) is 1/a ln|tan(ax)|+C

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integral of 1/(sin(ax)cos(ax))

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

\frac{1}{a} ln ∣ tan (a x) ∣ + C

Solution steps

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

Apply u-substitution

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

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

Rewrite using trig identities

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

Apply Trigonometric Substitution

: \int \frac{1}{v} d v

= \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 (a x) ∣

Add a constant to the solution

= \frac{1}{a} ln ∣ tan (a x) ∣ + C

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