What is the integral of x^2sin(θx) ?

The integral of x^2sin(θx) is -1/θ x^2cos(θx)+2/(θ^3)(θxsin(θx)+cos(θx))+C

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integral of x^2sin(θx)

\int x^{2} sin (θ x) d x

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

Solution steps

\int x^{2} sin (θ x) d x

Apply Integration By Parts

: - \frac{1}{θ} x^{2} cos (θ x) - \int - \frac{2}{θ} x cos (θ x) d x

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

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

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

Simplify

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

Add a constant to the solution

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

\int_{0}^{5} π (x - 1)^{2} d x

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

\int sin (π) x cos (π) x d x

x \to 5 lim (- x^{2} + x - 5)

\int \frac{3 x ^{2}}{x ^{2} + 1} d x

What is the integral of x^2sin(θx) ?
The integral of x^2sin(θx) is -1/θ x^2cos(θx)+2/(θ^3)(θxsin(θx)+cos(θx))+C