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

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

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integral of 2x^2sin(2x)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x^{2} sin (2 x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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