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

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

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

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

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

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

Using integration by parts one more time, with u = 6x and dv = cos(x) dx, we get:

∫6x*cos(x) dx = 6x*sin(x) + ∫6*sin(x) dx

Putting it all together, we have:

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

where C is the constant of integration.