What is the integral of (x^3+2x)sin(x) ?

The integral of (x^3+2x)sin(x) is -x^3cos(x)+3x^2sin(x)+4xcos(x)-4sin(x)+C

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integral of (x^3+2x)sin(x)

\int (x^{3} + 2 x) sin (x) d x

- x^{3} cos (x) + 3 x^{2} sin (x) + 4 x cos (x) - 4 sin (x) + C

Solution steps

\int (x^{3} + 2 x) sin (x) d x

Expand

(x^{3} + 2 x) sin (x) : x^{3} sin (x) + 2 x sin (x)

= \int x^{3} sin (x) + 2 x 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^{3} sin (x) d x + \int 2 x sin (x) d x

\int x^{3} sin (x) d x = - x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))

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

= - x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))) + 2 (- x cos (x) + sin (x))

Simplify

- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))) + 2 (- x cos (x) + sin (x)) : - x^{3} cos (x) + 3 x^{2} sin (x) + 4 x cos (x) - 4 sin (x)

= - x^{3} cos (x) + 3 x^{2} sin (x) + 4 x cos (x) - 4 sin (x)

Add a constant to the solution

= - x^{3} cos (x) + 3 x^{2} sin (x) + 4 x cos (x) - 4 sin (x) + C

What is the integral of (x^3+2x)sin(x) ?
The integral of (x^3+2x)sin(x) is -x^3cos(x)+3x^2sin(x)+4xcos(x)-4sin(x)+C