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

The integral of x^3*sin(3x) is -1/3 x^3cos(3x)+1/27 (9x^2sin(3x)-2(-3xcos(3x)+sin(3x)))+C

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integral of x^3*sin(3x)

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

- \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))) + C

Solution steps

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

Apply Integration By Parts

: - \frac{1}{3} x^{3} cos (3 x) - \int - x^{2} cos (3 x) d x

= - \frac{1}{3} x^{3} cos (3 x) - \int - x^{2} cos (3 x) d x

\int - x^{2} cos (3 x) d x = - \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))

= - \frac{1}{3} x^{3} cos (3 x) - (- \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))))

Simplify

= - \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))

Add a constant to the solution

= - \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))) + C

\int_{0}^{3} (5 x - x^{2}) - (2 x) d x

\frac{d ^{2}}{d x ^{2}} (- \frac{2 x ^{2}}{5 y ^{2}})

\frac{d y}{d x} = e^{6 y + 10 x}

\frac{d}{d x} (2 x - \frac{16}{x ^{2}})

n = 3 \sum \infty e^{7 - 6 n}

What is the integral of x^3*sin(3x) ?
The integral of x^3*sin(3x) is -1/3 x^3cos(3x)+1/27 (9x^2sin(3x)-2(-3xcos(3x)+sin(3x)))+C