What is the integral of 6x^2sin(2x) ?

The integral of 6x^2sin(2x) is 6(-1/2 x^2cos(2x)+1/4 (2xsin(2x)+cos(2x)))+C

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integral of 6x^2sin(2x)

\int 6 x^{2} sin (2 x) d x

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

Solution steps

\int 6 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

= 6 \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

= 6 (- \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))

= 6 (- \frac{1}{2} x^{2} cos (2 x) - (- \frac{1}{4} (2 x sin (2 x) + cos (2 x))))

Simplify

= 6 (- \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x)))

Add a constant to the solution

= 6 (- \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x))) + C

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What is the integral of 6x^2sin(2x) ?
The integral of 6x^2sin(2x) is 6(-1/2 x^2cos(2x)+1/4 (2xsin(2x)+cos(2x)))+C