What is the integral of x^2sin(βx) ?

The integral of x^2sin(βx) is -1/β x^2cos(βx)+2/(β^3)(βxsin(βx)+cos(βx))+C

integral of x^2sin(βx)

\int x^{2} sin (β x) d x

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

Solution steps

\int x^{2} sin (β x) d x

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of x^2sin(βx) ?
The integral of x^2sin(βx) is -1/β x^2cos(βx)+2/(β^3)(βxsin(βx)+cos(βx))+C