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

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

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integral of sin(x)x^3

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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