What is the integral of t^2sin(βt) ?

The integral of t^2sin(βt) is -1/β t^2cos(βt)+2/(β^3)(βtsin(βt)+cos(βt))+C

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integral of t^2sin(βt)

\int t^{2} sin (βt) d t

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

Solution steps

\int t^{2} sin (βt) d t

Apply Integration By Parts

: - \frac{1}{β} t^{2} cos (βt) - \int - \frac{2}{β} t cos (βt) d t

= - \frac{1}{β} t^{2} cos (βt) - \int - \frac{2}{β} t cos (βt) d t

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

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

Simplify

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

Add a constant to the solution

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

\int x (x^{2} + 3)^{100} d x

\frac{d}{d x} (x ln (1 - x))

n \to \infty lim (2^{n})

\frac{\partial}{\partial x} (e^{3 x})

x \to - 9 + lim (\frac{6 x + 54}{x + 9})

What is the integral of t^2sin(βt) ?
The integral of t^2sin(βt) is -1/β t^2cos(βt)+2/(β^3)(βtsin(βt)+cos(βt))+C