What is the integral of tan(xy) ?

The integral of tan(xy) is -1/y ln|cos(yx)|+C

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integral of tan(xy)

\int tan (x y) d x

- \frac{1}{y} ln ∣ cos (y x) ∣ + C

Solution steps

\int tan (x y) d x

Rewrite using trig identities

= \int \frac{sin ( y x )}{cos ( y x )} d x

Apply u-substitution

: \int - \frac{1}{y u} d u

= \int - \frac{1}{y u} d u

Take the constant out:

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

= - \frac{1}{y} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{y} ln ∣ u ∣

Substitute back

u = cos (y x)

= - \frac{1}{y} ln ∣ cos (y x) ∣

Add a constant to the solution

= - \frac{1}{y} ln ∣ cos (y x) ∣ + C

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