What is the integral of sin(3x)tan(3x) ?

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

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integral of sin(3x)tan(3x)

\int sin (3 x) tan (3 x) d x

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

Solution steps

\int sin (3 x) tan (3 x) d x

Apply Integration By Parts

= - \int - sec^{2} (3 x) cos (3 x) d x - \frac{1}{3} sin (3 x)

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

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

Simplify

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

Add a constant to the solution

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

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