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

The integral of x^3sin(1/3 x) is -3x^3cos(x/3)+9(3x^2sin(x/3)-6(-3xcos(x/3)+9sin(x/3)))+C

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

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

- 3 x^{3} cos (\frac{x}{3}) + 9 (3 x^{2} sin (\frac{x}{3}) - 6 (- 3 x cos (\frac{x}{3}) + 9 sin (\frac{x}{3}))) + C

Solution steps

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

Simplify

\frac{1}{3} x : \frac{x}{3}

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

Apply Integration By Parts

: - 3 x^{3} cos (\frac{x}{3}) - \int - 9 x^{2} cos (\frac{x}{3}) d x

= - 3 x^{3} cos (\frac{x}{3}) - \int - 9 x^{2} cos (\frac{x}{3}) d x

\int - 9 x^{2} cos (\frac{x}{3}) d x = - 9 (3 x^{2} sin (\frac{x}{3}) - 6 (- 3 x cos (\frac{x}{3}) + 9 sin (\frac{x}{3})))

= - 3 x^{3} cos (\frac{x}{3}) - (- 9 (3 x^{2} sin (\frac{x}{3}) - 6 (- 3 x cos (\frac{x}{3}) + 9 sin (\frac{x}{3}))))

Simplify

= - 3 x^{3} cos (\frac{x}{3}) + 9 (3 x^{2} sin (\frac{x}{3}) - 6 (- 3 x cos (\frac{x}{3}) + 9 sin (\frac{x}{3})))

Add a constant to the solution

= - 3 x^{3} cos (\frac{x}{3}) + 9 (3 x^{2} sin (\frac{x}{3}) - 6 (- 3 x cos (\frac{x}{3}) + 9 sin (\frac{x}{3}))) + C

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What is the integral of x^3sin(1/3 x) ?
The integral of x^3sin(1/3 x) is -3x^3cos(x/3)+9(3x^2sin(x/3)-6(-3xcos(x/3)+9sin(x/3)))+C