What is the integral of tan(z) ?

The integral of tan(z) is -ln|cos(z)|+C

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integral of tan(z)

\int tan (z) d z

- ln ∣ cos (z) ∣ + C

Solution steps

\int tan (z) d z

Rewrite using trig identities

= \int \frac{sin ( z )}{cos ( z )} d z

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = cos (z)

= - ln ∣ cos (z) ∣

Add a constant to the solution

= - ln ∣ cos (z) ∣ + C