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

The integral of (x+6)^2sin(x) is -x^2cos(x)+2xsin(x)-12xcos(x)-34cos(x)+12sin(x)+C

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integral of (x+6)^2sin(x)

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

- x^{2} cos (x) + 2 x sin (x) - 12 x cos (x) - 34 cos (x) + 12 sin (x) + C

Solution steps

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

x + 6 = 6 (\frac{x}{6} + 1)

= \int (6 (\frac{x}{6} + 1))^{2} sin (x) d x

Simplify

(6 (\frac{x}{6} + 1))^{2} sin (x) : x^{2} sin (x) + 12 x sin (x) + 36 sin (x)

= \int x^{2} sin (x) + 12 x sin (x) + 36 sin (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} sin (x) d x + \int 12 x sin (x) d x + \int 36 sin (x) d x

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

\int 12 x sin (x) d x = 12 (- x cos (x) + sin (x))

\int 36 sin (x) d x = - 36 cos (x)

= - x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 12 (- x cos (x) + sin (x)) - 36 cos (x)

Simplify

- x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 12 (- x cos (x) + sin (x)) - 36 cos (x) : - x^{2} cos (x) + 2 x sin (x) - 12 x cos (x) - 34 cos (x) + 12 sin (x)

= - x^{2} cos (x) + 2 x sin (x) - 12 x cos (x) - 34 cos (x) + 12 sin (x)

Add a constant to the solution

= - x^{2} cos (x) + 2 x sin (x) - 12 x cos (x) - 34 cos (x) + 12 sin (x) + C

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What is the integral of (x+6)^2sin(x) ?
The integral of (x+6)^2sin(x) is -x^2cos(x)+2xsin(x)-12xcos(x)-34cos(x)+12sin(x)+C