What is the integral of (x^2+5x+6)sin(x) ?

The integral of (x^2+5x+6)sin(x) is -x^2cos(x)+2xsin(x)-5xcos(x)-4cos(x)+5sin(x)+C

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integral of (x^2+5x+6)sin(x)

\int (x^{2} + 5 x + 6) sin (x) d x

- x^{2} cos (x) + 2 x sin (x) - 5 x cos (x) - 4 cos (x) + 5 sin (x) + C

Solution steps

\int (x^{2} + 5 x + 6) sin (x) d x

Expand

(x^{2} + 5 x + 6) sin (x) : x^{2} sin (x) + 5 x sin (x) + 6 sin (x)

= \int x^{2} sin (x) + 5 x sin (x) + 6 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 5 x sin (x) d x + \int 6 sin (x) d x

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

\int 5 x sin (x) d x = 5 (- x cos (x) + sin (x))

\int 6 sin (x) d x = - 6 cos (x)

= - x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 5 (- x cos (x) + sin (x)) - 6 cos (x)

Simplify

- x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 5 (- x cos (x) + sin (x)) - 6 cos (x) : - x^{2} cos (x) + 2 x sin (x) - 5 x cos (x) - 4 cos (x) + 5 sin (x)

= - x^{2} cos (x) + 2 x sin (x) - 5 x cos (x) - 4 cos (x) + 5 sin (x)

Add a constant to the solution

= - x^{2} cos (x) + 2 x sin (x) - 5 x cos (x) - 4 cos (x) + 5 sin (x) + C

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What is the integral of (x^2+5x+6)sin(x) ?
The integral of (x^2+5x+6)sin(x) is -x^2cos(x)+2xsin(x)-5xcos(x)-4cos(x)+5sin(x)+C