/*! 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 14 Find the indefinite integral. ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 14

Find the indefinite integral. $$ \int \frac{x(x-2)}{(x-1)^{3}} d x $$

Short Answer

Expert verified
\(\int \frac{x(x-2)}{(x-1)^{3}} dx = -\frac{1}{x-1} -\frac{1}{(x-1)^2} + C\)

Step by step solution

01

Simplifying the Function

It is essential to simplify the function before integrating it. The function given is of the form \(\frac{P(x)}{Q(x)}\), where both \(P\) and \(Q\) are polynomials. \(Q\) has higher degree than \(P\), so we can use polynomial division to simplify the function. In particular, we are able to rewrite \(x(x-2)\) as \(x^2-2x\), then divide \(x^2-2x\) by \((x-1)^{3}\). This allows us to rewrite \(x^2-2x\) as \((x-1)+(1)\) or \((x-1)\cdot1+1\). Then we find the remainder is 2. So, \(\frac{x(x-2)}{(x-1)^{3}} = \frac{1}{(x-1)^{2}} + \frac{2}{(x-1)^{3}}\).
02

Integrating the Simplified Function

We integrate the resulting expressions individually: first, \(\int \frac{1}{(x-1)^{2}} dx = -\frac{1}{x-1}\) and \(\int \frac{2}{(x-1)^{3}} dx = -\frac{1}{(x-1)^2}\). Thus, the integral in the original problem is equal to the sum of these two integrals.
03

Final Answer and Absolute Constant

Because we're solving an indefinite integral, we add a constant of integration at the end. Thus, our final answer is \(\int \frac{x(x-2)}{(x-1)^{3}} dx = -\frac{1}{x-1} -\frac{1}{(x-1)^2} + C\).

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Most popular questions from this chapter

(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{\pi / 4}^{x} \sec ^{2} t d t $$

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Prove that $$\int_{a}^{b} x^{2} d x=\frac{b^{3}-a^{3}}{3}$$

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