What is the integral of tan(7z) ?

The integral of tan(7z) is -1/7 ln|cos(7z)|+C

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integral of tan(7z)

\int tan (7 z) d z

- \frac{1}{7} ln ∣ cos (7 z) ∣ + C

Solution steps

\int tan (7 z) d z

Rewrite using trig identities

= \int \frac{sin ( 7 z )}{cos ( 7 z )} d z

Apply u-substitution

= \int - \frac{1}{7 u} d u

Take the constant out:

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

= - \frac{1}{7} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{7} ln ∣ u ∣

Substitute back

u = cos (7 z)

= - \frac{1}{7} ln ∣ cos (7 z) ∣

Add a constant to the solution

= - \frac{1}{7} ln ∣ cos (7 z) ∣ + C