What's the integral of x^2e^xcos(x)?

The integral of x^2e^xcos(x) is (x^2/2)e^xcos(x) + xe^xsin(x) - 2∫xe^xsin(x)dx.

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

∫x^2e^xcos(x)dx = x^2e^xsin(x) - ∫2xe^xsin(x)dx

Now we need to integrate ∫2xe^xsin(x)dx. Let u = x and dv = e^xsin(x)dx. Then du/dx = 1 and v = -e^xcos(x). Using integration by parts again, we have:

∫2xe^xsin(x)dx = -2xe^xcos(x) + 2∫e^xcos(x)dx

Now we can substitute this back into our original equation:

∫x^2e^xcos(x)dx = x^2e^xsin(x) - (-2xe^xcos(x) + 2∫e^xcos(x)dx)

Simplifying, we get:

∫x^2e^xcos(x)dx = (x^2/2)e^xcos(x) + xe^xsin(x) - 2∫e^xcos(x)dx

Finally, we need to integrate ∫e^xcos(x)dx. This integral does not have a simple closed-form solution, so we will leave it in this form. Therefore, the final answer is:

∫x^2e^xcos(x)dx = (x^2/2)e^xcos(x) + xe^xsin(x) - 2∫e^xcos(x)dx