/*! 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 25 Find or evaluate the integral. (... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 25

Find or evaluate the integral. (Complete the square, if necessary.) $$ \int \frac{2 x-5}{x^{2}+2 x+2} d x $$

Short Answer

Expert verified
The integral is \(\ln |(x+1)^{2}+1| - 5 \arctan(x+1) + C\).

Step by step solution

01

Rewrite the integral

Rewrite the integral as \(\int \frac{2x}{x^{2}+2x+2}dx - \int \frac{5}{x^{2}+2x+2}dx\) to divide the two terms to facilitate the next steps.
02

Complete the square in the denominator

Complete the square in the denominator of each fractional part to make it easier to integrate. The denominator \(x^{2} + 2x + 2\) can be written in the form \((x+1)^{2}+1\). Now our integral looks like this: \(\int \frac{2x}{(x+1)^{2}+1}dx - \int \frac{5}{(x+1)^{2}+1}dx\).
03

Find the integral of each term

Next, we integrate each term separately. For the first term: The substitution \(u = x+1\) simplifies the denominator to \(u^{2}+1\), this is recognized as the integral of tan, and results in \(\int \frac{2u}{u^{2}+1}du = \ln |u^{2}+1|\). For the second term: This turns out to be the integral form of arctan(x), resulting in \(\int \frac{5}{u^{2}+1}du = 5 \arctan(u)\). Be sure to change back the variable of integration from u to x in your final answer.
04

Combine the results

Final step is to combine the results from step 3. We get \(\ln |(x+1)^{2}+1| - 5\arctan(x+1) + C\), where \(C\) represents the constant of integration.

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Most popular questions from this chapter

Consider the integral \(\int \frac{1}{\sqrt{6 x-x^{2}}} d x\). (a) Find the integral by completing the square of the radicand. (b) Find the integral by making the substitution \(u=\sqrt{x}\). (c) The antiderivatives in parts (a) and (b) appear to be significantly different. Use a graphing utility to graph each antiderivative in the same viewing window and determine the relationship between them. Find the domain of each.

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