/*! 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 48 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 48

Evaluate the following integrals. $$\int \frac{2 x+1}{x^{2}+4} d x$$

Short Answer

Expert verified
Question: Evaluate the indefinite integral $$\int \frac{2x + 1}{x^2 + 4} d x$$ Answer: $$\frac{3}{2} \ln |x^2 + 4| + C$$

Step by step solution

01

Identify the substitution

Let's substitute \(x^2 + 4\) as \(u\). Then, differentiating both sides with respect to \(x\), we get the following relation: $$\frac{d u}{d x} = 2x$$ Now we can solve for \(\frac{d x}{d u}\) : $$\frac{d x}{d u} = \frac{1}{2x}$$
02

Rewrite the integral in terms of \(u\)

We have: $$\frac{2x + 1}{x^2 + 4} d x = \frac{2x + 1}{u} \cdot \frac{d x}{d u} d u$$ Substitute the expression of \(\frac{d x}{d u}\) that we found in Step 1: $$\frac{2x + 1}{u} \cdot \frac{1}{2x} d u$$ Now, the \(2x\) terms will cancel out: $$\frac{2x + 1}{u} \cdot \frac{1}{2x} d u = \frac{1}{u} d u + \frac{1}{2x} \cdot \frac{1}{u} d u = \frac{1}{u} d u + \frac{1}{2u} d u$$
03

Evaluate the integrals

Now we can integrate these two terms: $$\int \frac{1}{u} d u + \frac{1}{2} \int \frac{1}{u} d u$$ The integral of \(\frac{1}{u}\) with respect to \(u\) is \(\ln |u|\). So, our expression becomes: $$\int \frac{1}{u} d u + \frac{1}{2} \int \frac{1}{u} d u = \ln |u| + \frac{1}{2} \ln |u|$$
04

Substitute back for \(x\)

Now, we can substitute back for \(u\) using our initial substitution relationship \(u = x^2 + 4\). $$\ln |x^2 + 4| + \frac{1}{2} \ln |x^2 + 4| = \frac{3}{2} \ln |x^2 + 4|$$ As an indefinite integral, don't forget to add the constant of integration \(C\): $$\frac{3}{2} \ln |x^2 + 4| + C$$ So, the final answer is: $$\int \frac{2x + 1}{x^2 + 4} d x = \frac{3}{2} \ln |x^2 + 4| + C$$

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Most popular questions from this chapter

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