What is the integral of (2tan(3x)-sec(2x)) ?

The integral of (2tan(3x)-sec(2x)) is -2/3 ln|cos(3x)|-1/2 ln|tan(2x)+sec(2x)|+C

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integral of (2tan(3x)-sec(2x))

\int (2 tan (3 x) - sec (2 x)) d x

- \frac{2}{3} ln ∣ cos (3 x) ∣ - \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣ + C

Solution steps

\int 2 tan (3 x) - sec (2 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 2 tan (3 x) d x - \int sec (2 x) d x

\int 2 tan (3 x) d x = - \frac{2}{3} ln ∣ cos (3 x) ∣

\int sec (2 x) d x = \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣

= - \frac{2}{3} ln ∣ cos (3 x) ∣ - \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣

Add a constant to the solution

= - \frac{2}{3} ln ∣ cos (3 x) ∣ - \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣ + C

What is the integral of (2tan(3x)-sec(2x)) ?
The integral of (2tan(3x)-sec(2x)) is -2/3 ln|cos(3x)|-1/2 ln|tan(2x)+sec(2x)|+C