What is the integral of-12tan(4t) ?

The integral of-12tan(4t) is 3ln|cos(4t)|+C

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integral of-12tan(4t)

\int - 12 tan (4 t) d t

3 ln ∣ cos (4 t) ∣ + C

Solution steps

\int - 12 tan (4 t) d t

Take the constant out:

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

= - 12 \cdot \int tan (4 t) d t

Rewrite using trig identities

= - 12 \cdot \int \frac{sin ( 4 t )}{cos ( 4 t )} d t

Apply u-substitution

= - 12 \cdot \int - \frac{1}{4 u} d u

Take the constant out:

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

= - 12 (- \frac{1}{4} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 12 (- \frac{1}{4} ln ∣ u ∣)

Substitute back

u = cos (4 t)

= - 12 (- \frac{1}{4} ln ∣ cos (4 t) ∣)

Simplify

- 12 (- \frac{1}{4} ln ∣ cos (4 t) ∣) : 3 ln ∣ cos (4 t) ∣

= 3 ln ∣ cos (4 t) ∣

Add a constant to the solution

= 3 ln ∣ cos (4 t) ∣ + C