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

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

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integral of x^3*sin(x)

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

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

Solution steps

\int x^{3} sin (x) 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

\int_{0}^{2} x ln (x) d x

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\frac{d}{d x} (lo g_{e} (sin (x)))

x + 2 y = 9, x + 6 = y^{2}

\int sin^{5} (13 x) d x

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