What is the integral of x^2sin(kx) ?

The integral of x^2sin(kx) is -1/k x^2cos(kx)+2/(k^3)(kxsin(kx)+cos(kx))+C

integral of x^2sin(kx)

\int x^{2} sin (k x) d x

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

Solution steps

\int x^{2} sin (k x) d x

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of x^2sin(kx) ?
The integral of x^2sin(kx) is -1/k x^2cos(kx)+2/(k^3)(kxsin(kx)+cos(kx))+C