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

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

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integral of sin(3t)tan(3t)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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