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

The integral of t^2cos(3t) is 1/27 (9t^2sin(3t)-2(-3tcos(3t)+sin(3t)))+C

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integral of t^2cos(3t)

\int t^{2} cos (3 t) d t

\frac{1}{27} (9 t^{2} sin (3 t) - 2 (- 3 t cos (3 t) + sin (3 t))) + C

Solution steps

\int t^{2} cos (3 t) d t

Apply u-substitution

: \int \frac{u ^{2} cos ( u )}{27} d u

= \int \frac{u ^{2} cos ( u )}{27} d u

Take the constant out:

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

= \frac{1}{27} \cdot \int u^{2} cos (u) d u

Apply Integration By Parts

: u^{2} sin (u) - \int 2 u sin (u) d u

= \frac{1}{27} (u^{2} sin (u) - \int 2 u sin (u) d u)

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

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

Substitute back

u = 3 t

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

Simplify

\frac{1}{27} ((3 t)^{2} sin (3 t) - 2 (- 3 t cos (3 t) + sin (3 t))) : \frac{1}{27} (9 t^{2} sin (3 t) - 2 (- 3 t cos (3 t) + sin (3 t)))

= \frac{1}{27} (9 t^{2} sin (3 t) - 2 (- 3 t cos (3 t) + sin (3 t)))

Add a constant to the solution

= \frac{1}{27} (9 t^{2} sin (3 t) - 2 (- 3 t cos (3 t) + sin (3 t))) + C

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What is the integral of t^2cos(3t) ?
The integral of t^2cos(3t) is 1/27 (9t^2sin(3t)-2(-3tcos(3t)+sin(3t)))+C