/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 34

Evaluate the following integrals. $$\int \frac{x^{2}}{(x-2)^{3}} d x$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int \frac{x^{2}}{(x-2)^{3}} d x$$. Answer: $$\int \frac{x^{2}}{(x-2)^{3}} d x = \ln |x-2| - \frac{4}{x-2} + \frac{2}{(x-2)^2} + C$$

Step by step solution

01

Choosing the right substitution

Let's choose the substitution u to simplify the given function: $$u = x - 2$$ Now, find the derivative of u with respect to x: $$\frac{d u}{d x} = 1$$ Now, $$dx = du$$
02

Substitute and Simplify

Replace x and dx in the given integral with their corresponding u and du: $$\int \frac{(u+2)^{2}}{u^{3}} d u$$ Now, expand the numerator: $$\int \frac{u^2 + 4u + 4}{u^{3}} du$$ Now, simplify the integral by breaking it into separate fractions: $$\int \left(\frac{u^2}{u^3} + \frac{4u}{u^3} + \frac{4}{u^3}\right) du$$ Simplify further: $$\int \left(\frac{1}{u} + \frac{4}{u^2} + \frac{4}{u^3}\right) du$$
03

Evaluate the Integral

Now, evaluate the integral of each term: $$\int \left(\frac{1}{u} + \frac{4}{u^2} + \frac{4}{u^3}\right) du = \int \frac{1}{u} \, du + \int \frac{4}{u^2} \, du + \int \frac{4}{u^3} \, du$$ Integration of the first term: $$\int \frac{1}{u} \, du = \ln |u| + C_1$$ Integration of the second term: $$\int \frac{4}{u^2} \, du = -\frac{4}{u} + C_2$$ Integration of the third term: $$\int \frac{4}{u^3} \, du = \frac{2}{u^2} + C_3$$
04

Combine the Results and Substitute Back

Now, we will combine our results: $$\int \left(\frac{1}{u} + \frac{4}{u^2} + \frac{4}{u^3}\right) du = \ln |u| - \frac{4}{u} + \frac{2}{u^2} + C$$ where $$C = C_1 + C_2 + C_3$$ Now substitute back for u: $$= \ln |x-2| - \frac{4}{x-2} + \frac{2}{(x-2)^2} + C$$ So the final result is: $$\int \frac{x^{2}}{(x-2)^{3}} d x = \ln |x-2| - \frac{4}{x-2} + \frac{2}{(x-2)^2} + C$$

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