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

The integral of x^2sin(-6x) is 1/6 x^2cos(6x)-1/18 xsin(6x)-1/108 cos(6x)+C

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integral of x^2sin(-6x)

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

\frac{1}{6} x^{2} cos (6 x) - \frac{1}{18} x sin (6 x) - \frac{1}{108} cos (6 x) + C

Solution steps

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

Simplify

x^{2} sin (- 6 x) : - x^{2} sin (6 x)

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

Take the constant out:

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

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

Apply Integration By Parts

: - \frac{1}{6} x^{2} cos (6 x) - \int - \frac{1}{3} x cos (6 x) d x

= - (- \frac{1}{6} x^{2} cos (6 x) - \int - \frac{1}{3} x cos (6 x) d x)

\int - \frac{1}{3} x cos (6 x) d x = - \frac{1}{108} (6 x sin (6 x) + cos (6 x))

= - (- \frac{1}{6} x^{2} cos (6 x) - (- \frac{1}{108} (6 x sin (6 x) + cos (6 x))))

Simplify

- (- \frac{1}{6} x^{2} cos (6 x) - (- \frac{1}{108} (6 x sin (6 x) + cos (6 x)))) : \frac{1}{6} x^{2} cos (6 x) - \frac{1}{18} x sin (6 x) - \frac{1}{108} cos (6 x)

= \frac{1}{6} x^{2} cos (6 x) - \frac{1}{18} x sin (6 x) - \frac{1}{108} cos (6 x)

Add a constant to the solution

= \frac{1}{6} x^{2} cos (6 x) - \frac{1}{18} x sin (6 x) - \frac{1}{108} cos (6 x) + C

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What is the integral of x^2sin(-6x) ?
The integral of x^2sin(-6x) is 1/6 x^2cos(6x)-1/18 xsin(6x)-1/108 cos(6x)+C