What is the integral of (sin(3x)-tan(2x)+5/(3x+2)) ?

The integral of (sin(3x)-tan(2x)+5/(3x+2)) is -1/3 cos(3x)+1/2 ln|cos(2x)|+5/3 ln|3x+2|+C

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integral of (sin(3x)-tan(2x)+5/(3x+2))

\int (sin (3 x) - tan (2 x) + \frac{5}{3 x + 2}) d x

- \frac{1}{3} cos (3 x) + \frac{1}{2} ln ∣ cos (2 x) ∣ + \frac{5}{3} ln ∣ 3 x + 2 ∣ + C

Solution steps

\int sin (3 x) - tan (2 x) + \frac{5}{3 x + 2} 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 sin (3 x) d x - \int tan (2 x) d x + \int \frac{5}{3 x + 2} d x

\int sin (3 x) d x = - \frac{1}{3} cos (3 x)

\int tan (2 x) d x = - \frac{1}{2} ln ∣ cos (2 x) ∣

\int \frac{5}{3 x + 2} d x = \frac{5}{3} ln ∣ 3 x + 2 ∣

= - \frac{1}{3} cos (3 x) - (- \frac{1}{2} ln ∣ cos (2 x) ∣) + \frac{5}{3} ln ∣ 3 x + 2 ∣

Simplify

= - \frac{1}{3} cos (3 x) + \frac{1}{2} ln ∣ cos (2 x) ∣ + \frac{5}{3} ln ∣ 3 x + 2 ∣

Add a constant to the solution

= - \frac{1}{3} cos (3 x) + \frac{1}{2} ln ∣ cos (2 x) ∣ + \frac{5}{3} ln ∣ 3 x + 2 ∣ + C

What is the integral of (sin(3x)-tan(2x)+5/(3x+2)) ?
The integral of (sin(3x)-tan(2x)+5/(3x+2)) is -1/3 cos(3x)+1/2 ln|cos(2x)|+5/3 ln|3x+2|+C