What is the integral of (x^2)sin(4x) ?

The integral of (x^2)sin(4x) is -1/4 x^2cos(4x)+1/32 (4xsin(4x)+cos(4x))+C

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integral of (x^2)sin(4x)

\int (x^{2}) sin (4 x) d x

- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)) + C

Solution steps

\int x^{2} sin (4 x) d x

Apply Integration By Parts

: - \frac{1}{4} x^{2} cos (4 x) - \int - \frac{1}{2} x cos (4 x) d x

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

\int - \frac{1}{2} x cos (4 x) d x = - \frac{1}{32} (4 x sin (4 x) + cos (4 x))

= - \frac{1}{4} x^{2} cos (4 x) - (- \frac{1}{32} (4 x sin (4 x) + cos (4 x)))

Simplify

= - \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x))

Add a constant to the solution

= - \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)) + C

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What is the integral of (x^2)sin(4x) ?
The integral of (x^2)sin(4x) is -1/4 x^2cos(4x)+1/32 (4xsin(4x)+cos(4x))+C