What is the integral of tan(3t) ?

The integral of tan(3t) is -1/3 ln|cos(3t)|+C

Popular >

integral of tan(3t)

\int tan (3 t) d t

- \frac{1}{3} ln ∣ cos (3 t) ∣ + C

Solution steps

\int tan (3 t) d t

Rewrite using trig identities

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

Apply u-substitution

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

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{3} ln ∣ u ∣

Substitute back

u = cos (3 t)

= - \frac{1}{3} ln ∣ cos (3 t) ∣

Add a constant to the solution

= - \frac{1}{3} ln ∣ cos (3 t) ∣ + C

y^{^{'''}} + y^{^{''}} = x

\int \frac{3}{3 + e ^{x}} d x

y^{^{''}} + y^{^{'}} - 30 y = 0, y (0) = 2, y^{^{'}} (0) = - 2

(2 x + 1)^{^{'}}

y^{^{'}} = (y + 2 x)^{2}, y (0) = 3