What is the integral of (x^3+1)sin(x) ?

The integral of (x^3+1)sin(x) is -x^3cos(x)+3(x^2sin(x)-2(-xcos(x)+sin(x)))-cos(x)+C

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integral of (x^3+1)sin(x)

\int (x^{3} + 1) sin (x) d x

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

Solution steps

\int (x^{3} + 1) sin (x) d x

Expand

(x^{3} + 1) sin (x) : x^{3} sin (x) + sin (x)

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

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

Add a constant to the solution

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

\int x (3 x - 2)^{2} d x

f (x) = \frac{x ^{2} + 3 x}{6}

\int (6 x^{6} - 6 x^{5} - 4 x^{3} + 2) d x

ln (cos (x)), 0

\frac{d}{d x} (\frac{9}{x ^{2} + 5})

What is the integral of (x^3+1)sin(x) ?
The integral of (x^3+1)sin(x) is -x^3cos(x)+3(x^2sin(x)-2(-xcos(x)+sin(x)))-cos(x)+C