What is the result of the integral ∫ x e^x dx using integration by parts?

To find the integral ∫ x e^x dx using integration by parts, we utilize the formula for integration by parts, which is given by:

∫ u dv = uv - ∫ v du.

We need to choose functions for u and dv. Typically, in integration by parts, we let:

• u = x (which makes du = dx),

• dv = e^x dx (which gives v = e^x).

Now we apply the integration by parts formula:

1. Substitute our choices into the formula:

∫ x e^x dx = x e^x - ∫ e^x dx.

2. Next, we need to evaluate the integral ∫ e^x dx, which results in e^x. Therefore,

∫ x e^x dx = x e^x - e^x + C.

3. This can be simplified to:

∫ x e^x dx = (x - 1)e^x + C.

This matches the form given in the first option. The constant C indicates the integration constant, which is always included when performing indefinite integrals. Thus, the result for the integral ∫ x e^x dx using integration by parts is indeed