Final answer:
To solve ∫ x^2/(x-2)^3 dx, use substitution to divide the integral into simpler fractions, integrate each term, and sum them with the constant of integration. Your final solution will look something like the following (substituting u back to x - 2):∫ x^2/(x-2)^3 dx = ln|x - 2| - 4/(x - 2) - 2/(x - 2)^2 + C
Step-by-step explanation:
To evaluate the integral ∫ x^2/(x-2)^3 dx, you can use a substitution method. First, let u = x - 2, which implies that du = dx. Now the integral becomes ∫ (u+2)^2/u^3 du. Next, expand the numerator to get ∫ (u^2 + 4u + 4)/u^3 du and then divide each term by u^3 to separate into partial fractions which can be integrated term by term. Your new integral becomes ∫ u^{-1} + 4u^{-2} + 4u^{-3} du, or ∫ (1/u + 4/u^2 + 4/u^3) du.
Now perform the integration of each term separately: ∫ 1/u du = ln|u|, ∫ 4/u^2 du = -4/u, and ∫ 4/u^3 du = -2/u^2. After finding the integrals of these individual terms, you should sum them up and replace u with x - 2 to return to the original variable. Don't forget to add the constant of integration C at the end.
Your final solution will look something like the following (substituting u back to x - 2):
∫ x^2/(x-2)^3 dx = ln|x - 2| - 4/(x - 2) - 2/(x - 2)^2 + C