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

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

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integral of 2x^3sin(x)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

= 2 (- 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)))

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

Simplify

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

Add a constant to the solution

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