What is the integral of tan(6θ) ?

The integral of tan(6θ) is -1/6 ln|cos(6θ)|+C

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integral of tan(6θ)

\int tan (6 θ) d θ

- \frac{1}{6} ln ∣ cos (6 θ) ∣ + C

Solution steps

\int tan (6 θ) d θ

Rewrite using trig identities

= \int \frac{sin ( 6 θ )}{cos ( 6 θ )} d θ

Apply u-substitution

: \int - \frac{1}{6 u} d u

= \int - \frac{1}{6 u} d u

Take the constant out:

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

= - \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{6} ln ∣ u ∣

Substitute back

u = cos (6 θ)

= - \frac{1}{6} ln ∣ cos (6 θ) ∣

Add a constant to the solution

= - \frac{1}{6} ln ∣ cos (6 θ) ∣ + C

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