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

The integral of x^3*sin(2x) is -1/2 x^3cos(2x)+3/8 (2x^2sin(2x)+2xcos(2x)-sin(2x))+C

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integral of x^3*sin(2x)

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

- \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{3}{2} x^{2} cos (2 x) d x = - \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))

= - \frac{1}{2} x^{3} cos (2 x) - (- \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)))

Simplify

= - \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))

Add a constant to the solution

= - \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)) + C

y = \frac{5}{x ^{4}} - \frac{7}{x}

\frac{3 s}{( s - 1 ) ( s ^{2} + 1 )}

x \to \infty lim (2 x - 2)

\frac{1}{4 s + 1}

y^{^{'}} + 2 y = 6, y (0) = 0

What is the integral of x^3*sin(2x) ?
The integral of x^3*sin(2x) is -1/2 x^3cos(2x)+3/8 (2x^2sin(2x)+2xcos(2x)-sin(2x))+C