How do you evaluate the integral ∫ x^2 e^x dx using integration by parts?

To evaluate the integral ∫ x^2 e^x dx using integration by parts, we start by setting up the necessary components of the integration by parts formula, which is ∫ u dv = uv - ∫ v du.

For this integral, we typically choose:

• u = x^2, which implies that du = 2x dx.

• dv = e^x dx, so v = e^x.

Now we can substitute these into the integration by parts formula:

∫ x^2 e^x dx = x^2 e^x - ∫ e^x (2x) dx.

Next, we need to evaluate the integral ∫ e^x (2x) dx. We again apply integration by parts. For this integral:

• Let u = 2x, which gives du = 2 dx.

• Let dv = e^x dx, so v = e^x.

Using the integration by parts formula again:

∫ 2x e^x dx = 2x e^x - ∫ e^x (2) dx

= 2x e^x - 2 e^x.

Putting this result back into our previous expression gives: