What is the integral of 3tan(3x) ?

The integral of 3tan(3x) is -ln|cos(3x)|+C

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integral of 3tan(3x)

\int 3 tan (3 x) d x

- ln ∣ cos (3 x) ∣ + C

Solution steps

\int 3 tan (3 x) d x

Take the constant out:

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

= 3 \cdot \int tan (3 x) d x

Rewrite using trig identities

= 3 \cdot \int \frac{sin ( 3 x )}{cos ( 3 x )} d x

Apply u-substitution

: \int - \frac{1}{3 u} d u

= 3 \cdot \int - \frac{1}{3 u} d u

Take the constant out:

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

= 3 (- \frac{1}{3} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 (- \frac{1}{3} ln ∣ u ∣)

Substitute back

u = cos (3 x)

= 3 (- \frac{1}{3} ln ∣ cos (3 x) ∣)

Simplify

3 (- \frac{1}{3} ln ∣ cos (3 x) ∣) : - ln ∣ cos (3 x) ∣

= - ln ∣ cos (3 x) ∣

Add a constant to the solution

= - ln ∣ cos (3 x) ∣ + C

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