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

The integral of x^2sin(2Ax) is -1/(2A)x^2cos(2Ax)+1/A (1/(2A)xsin(2Ax)+1/(4A^2)cos(2Ax))+C

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integral of x^2sin(2Ax)

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

- \frac{1}{2 A} x^{2} cos (2 A x) + \frac{1}{A} (\frac{1}{2 A} x sin (2 A x) + \frac{1}{4 A ^{2}} cos (2 A x)) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{1}{A} x cos (2 A x) d x = - \frac{1}{A} (\frac{1}{2 A} x sin (2 A x) + \frac{1}{4 A ^{2}} cos (2 A x))

= - \frac{1}{2 A} x^{2} cos (2 A x) - (- \frac{1}{A} (\frac{1}{2 A} x sin (2 A x) + \frac{1}{4 A ^{2}} cos (2 A x)))

Simplify

= - \frac{1}{2 A} x^{2} cos (2 A x) + \frac{1}{A} (\frac{1}{2 A} x sin (2 A x) + \frac{1}{4 A ^{2}} cos (2 A x))

Add a constant to the solution

= - \frac{1}{2 A} x^{2} cos (2 A x) + \frac{1}{A} (\frac{1}{2 A} x sin (2 A x) + \frac{1}{4 A ^{2}} cos (2 A x)) + C

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What is the integral of x^2sin(2Ax) ?
The integral of x^2sin(2Ax) is -1/(2A)x^2cos(2Ax)+1/A (1/(2A)xsin(2Ax)+1/(4A^2)cos(2Ax))+C