What is the integral of (3t+t^2)sin(3t) ?

The integral of (3t+t^2)sin(3t) is -tcos(3t)+1/3 sin(3t)-1/3 t^2cos(3t)+2/27 (3tsin(3t)+cos(3t))+C

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integral of (3t+t^2)sin(3t)

\int (3 t + t^{2}) sin (3 t) d t

- t cos (3 t) + \frac{1}{3} sin (3 t) - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t)) + C

Solution steps

\int (3 t + t^{2}) sin (3 t) d t

Expand

(3 t + t^{2}) sin (3 t) : 3 t sin (3 t) + t^{2} sin (3 t)

= \int 3 t sin (3 t) + t^{2} sin (3 t) d t

Apply the Sum Rule:

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

= \int 3 t sin (3 t) d t + \int t^{2} sin (3 t) d t

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

\int t^{2} sin (3 t) d t = - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t))

= - t cos (3 t) + \frac{1}{3} sin (3 t) - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t))

Add a constant to the solution

= - t cos (3 t) + \frac{1}{3} sin (3 t) - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t)) + C

\int ln (1 + t^{2}) d t

d er i v a t i v e 2 w^{3 π}, at 7

in v erse l a pl a ce \frac{k}{s}

\frac{d}{d x} (cos^{2} (9 x) + sin^{2} (9 x))

\frac{d y}{d x} + \frac{y}{x ^{2} + x} = x + 1

What is the integral of (3t+t^2)sin(3t) ?
The integral of (3t+t^2)sin(3t) is -tcos(3t)+1/3 sin(3t)-1/3 t^2cos(3t)+2/27 (3tsin(3t)+cos(3t))+C