Mathematics

This problem is about integration by parts, a technique for handling products of polynomials and exponentials. The key insight is that you'll need to apply the integration-by-parts formula repeatedly—each time you integrate, the polynomial power drops by one, until you're left with just $e^x$. Part (b) then gives you a built-in check: differentiation should undo your work and return you to the original integrand, which is a powerful way to catch errors.

This problem is about integration by parts, a technique for handling products of polynomials and exponentials. The key insight is that you'll need to apply the integration-by-parts formula repeatedly—each time you integrate, the polynomial power drops by one, until you're left with just $e^x$. Part (b) then gives you a built-in check: differentiation should undo your work and return you to the original integrand, which is a powerful way to catch errors.