The integral ∫e^(-x) * x^(n-1) dx can be solved using integration by parts. Integration by parts is a technique that allows us to integrate a product of two functions by applying the formula:

∫u * v dx = u * ∫v dx - ∫u' * (∫v dx) dx

where u and v are functions of x, u' is the derivative of u with respect to x, and ∫v dx is the integral of v with respect to x.

To solve the given integral, let's choose u = x^(n-1) and dv = e^(-x) dx. This choice allows us to simplify the integral by differentiating u and integrating dv.

Integration by parts:
Applying the integration by parts formula to the given integral, we have:

∫e^(-x) * x^(n-1) dx = x^(n-1) * ∫e^(-x) dx - ∫(x^(n-1))' * (∫e^(-x) dx) dx

Simplifying the above equation, we get:

∫e^(-x) * x^(n-1) dx = x^(n-1) * (-e^(-x)) - ∫(n-1) * x^(n-2) * (-e^(-x)) dx

Solving the integral:
Now, we can simplify the above equation further by expanding and rearranging terms:

∫e^(-x) * x^(n-1) dx = -x^(n-1) * e^(-x) + (n-1) * ∫x^(n-2) * e^(-x) dx

We can continue applying integration by parts to the remaining integral on the right-hand side until we reach a base case. The base case occurs when n-2 = 0, which means n = 2.

Applying integration by parts again:
Using integration by parts on the remaining integral, we have:

∫x^(n-2) * e^(-x) dx = x^(n-2) * (-e^(-x)) + (n-2) * ∫x^(n-3) * (-e^(-x)) dx

Simplifying the above equation, we get:

∫x^(n-2) * e^(-x) dx = -x^(n-2) * e^(-x) - (n-2) * ∫x^(n-3) * e^(-x) dx

We can continue applying integration by parts in a similar manner until we reach the base case, which occurs when n-3 = 0, implying n = 3.

Base case:
When n = 2, the integral becomes:

∫e^(-x) * x dx = -x * e^(-x) + ∫(1) * e^(-x) dx

Simplifying further:

∫e^(-x) * x dx = -x * e^(-x) - e^(-x) + C

where C is the constant of integration.

Therefore, the solution to the given integral is:

∫e^(-x) * x^(n-1) dx = -x