What's the integral of x^4*cos(x)?

The integral of x^4*cos(x) is x^4*sin(x) + 4x^3*cos(x) + 12x^2*sin(x) - 24x*cos(x) - 24*sin(x) + C.

To solve this integral, we will use integration by parts. Let u = x^4 and dv = cos(x) dx. Then du/dx = 4x^3 and v = sin(x). Using the formula for integration by parts, we have:

∫ x^4*cos(x) dx = x^4*sin(x) - ∫ 4x^3*sin(x) dx

Next, we use integration by parts again, with u = 4x^3 and dv = sin(x) dx. Then du/dx = 12x^2 and v = -cos(x). Plugging these values into the formula, we get:

∫ x^4*cos(x) dx = x^4*sin(x) + 4x^3*cos(x) - ∫ 12x^2*cos(x) dx

We can continue this process, using integration by parts with u = 12x^2 and dv = -cos(x) dx. Then du/dx = 24x and v = -sin(x). Plugging these values into the formula, we get:

∫ x^4*cos(x) dx = x^4*sin(x) + 4x^3*cos(x) + 12x^2*sin(x) - ∫ 24x*sin(x) dx

Finally, we use integration by parts one last time, with u = 24x and dv = sin(x) dx. Then du/dx = 24 and v = -cos(x). Plugging these values into the formula, we get:

∫ x^4*cos(x) dx = x^4*sin(x) + 4x^3*cos(x) + 12x^2*sin(x) - 24x*cos(x) - ∫ 24*sin(x) dx

Simplifying the last integral, we get:

∫ x^4*cos(x) dx = x^4*sin(x) + 4x^3*cos(x) + 12x^2*sin(x) - 24x*cos(x) - 24*sin(x) + C

where C is the constant of integration.