Final answer:
The indefinite integral ∫(x-1)^2 e^x dx is solved using the method of integration by parts, by setting u = (x-1)^2 and dv = e^x dx. After applying the integration by parts formula, another simpler integral must be evaluated, to which the constant of integration will be added.
Step-by-step explanation:
To evaluate the indefinite integral ∫(x-1)^2 e^x dx, we can use the method of integration by parts. Integration by parts is based on the product rule for differentiation and is given by ∫ u dv = uv - ∫ v du, where u and dv are different parts of the integrand. We must choose 'u' and 'dv' such that the differentiation and integration processes simplify the problem.
Let u = (x-1)^2, which implies du = 2(x-1)dx. Next, we let dv = e^x dx, which gives v = e^x, the integral of dv. Applying integration by parts formula, we get:
∫(x-1)^2 e^x dx = u × v - ∫ v × du
= (x-1)^2 e^x - ∫ 2(x-1)e^x dx
Now, we have a simpler integral to evaluate, which may require another application of the integration by parts method or a different approach. Once we solve the integral, we add the constant of integration since it's an indefinite integral.
It is important to express your answers to problems to the correct number of significant figures and proper units.
Your complete question is: Evaluate the indefinite integral: ∫(x−1)^2 e^x dx