What is the integral of 27tsin(3t) ?

The integral of 27tsin(3t) is 27(-1/3 tcos(3t)+1/9 sin(3t))+C

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integral of 27tsin(3t)

\int 27 t sin (3 t) d t

27 (- \frac{1}{3} t cos (3 t) + \frac{1}{9} sin (3 t)) + C

Solution steps

\int 27 t sin (3 t) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 27 \cdot \int t sin (3 t) d t

Apply Integration By Parts

: - \frac{1}{3} t cos (3 t) - \int - \frac{1}{3} cos (3 t) d t

= 27 (- \frac{1}{3} t cos (3 t) - \int - \frac{1}{3} cos (3 t) d t)

\int - \frac{1}{3} cos (3 t) d t = - \frac{1}{9} sin (3 t)

= 27 (- \frac{1}{3} t cos (3 t) - (- \frac{1}{9} sin (3 t)))

Simplify

= 27 (- \frac{1}{3} t cos (3 t) + \frac{1}{9} sin (3 t))

Add a constant to the solution

= 27 (- \frac{1}{3} t cos (3 t) + \frac{1}{9} sin (3 t)) + C

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What is the integral of 27tsin(3t) ?
The integral of 27tsin(3t) is 27(-1/3 tcos(3t)+1/9 sin(3t))+C