To solve the integral [tex]\(\int \left(\frac{2x + 1}{x}\right) dx\)[/tex], we can follow these steps:
1. Break down the integrand:
The integrand [tex]\(\frac{2x + 1}{x}\)[/tex] can be simplified by splitting the fraction:
[tex]\[
\frac{2x + 1}{x} = \frac{2x}{x} + \frac{1}{x}
\][/tex]
2. Simplify the fractions separately:
[tex]\[
\frac{2x}{x} = 2 \quad \text{and} \quad \frac{1}{x} = \frac{1}{x}
\][/tex]
Therefore, our integrand becomes:
[tex]\[
\frac{2x + 1}{x} = 2 + \frac{1}{x}
\][/tex]
3. Set up the integral with the simplified form:
[tex]\[
\int \left(2 + \frac{1}{x}\right) dx
\][/tex]
4. Integrate term by term:
[tex]\[
\int 2 dx + \int \frac{1}{x} dx
\][/tex]
- The integral of [tex]\(2\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
\int 2 dx = 2x
\][/tex]
- The integral of [tex]\(\frac{1}{x}\)[/tex] with respect to [tex]\(x\)[/tex] is a standard integral, yielding:
[tex]\[
\int \frac{1}{x} dx = \ln|x|
\][/tex]
5. Combine the results:
Putting these together, we have:
[tex]\[
2x + \ln|x| + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
Thus, the solution to the integral [tex]\(\int \left( \frac{2x + 1}{x} \right) dx\)[/tex] is:
[tex]\[
2x + \ln|x| + C
\][/tex]
Given the conventions and the problem likely involving real numbers, we typically write the solution as:
[tex]\[
2x + \ln(x) + C
\][/tex]
unless [tex]\(x\)[/tex] is known to potentially be negative.
