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

The integral of x^3*sin(x) is -x^3*cos(x) + 3x^2*sin(x) + 6x*cos(x) - 6sin(x) + C.

To solve this integral, we can use integration by parts. Let u = x^3 and dv = sin(x) dx. Then du/dx = 3x^2 and v = -cos(x). Using the formula for integration by parts, we have:

∫x^3*sin(x) dx = -x^3*cos(x) - ∫-cos(x)*3x^2 dx

Next, we can use integration by parts again with u = 3x^2 and dv = -cos(x) dx. Then du/dx = 6x and v = sin(x). Plugging this into the formula for integration by parts, we get:

∫x^3*sin(x) dx = -x^3*cos(x) - (-3x^2*sin(x) - ∫-sin(x)*6x dx)

Simplifying this expression, we get:

∫x^3*sin(x) dx = -x^3*cos(x) + 3x^2*sin(x) + 6x*cos(x) - 6sin(x) + C

where C is the constant of integration. Therefore, the integral of x^3*sin(x) is -x^3*cos(x) + 3x^2*sin(x) + 6x*cos(x) - 6sin(x) + C.