What is the integral of tan(4t) ?

The integral of tan(4t) is -1/4 ln|cos(4t)|+C

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integral of tan(4t)

\int tan (4 t) d t

- \frac{1}{4} ln ∣ cos (4 t) ∣ + C

Solution steps

\int tan (4 t) d t

Rewrite using trig identities

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

Apply u-substitution

: \int - \frac{1}{4 u} d u

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{4} ln ∣ u ∣

Substitute back

u = cos (4 t)

= - \frac{1}{4} ln ∣ cos (4 t) ∣

Add a constant to the solution

= - \frac{1}{4} ln ∣ cos (4 t) ∣ + C

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