To evaluate the integral [tex]\(\int \frac{(2x + 3)(3x - 2)}{x^2} \, dx\)[/tex], let's go through the steps:
1. Expand the Numerator: First, we expand the expression in the numerator [tex]\((2x + 3)(3x - 2)\)[/tex].
[tex]\[
(2x + 3)(3x - 2) = 2x \cdot 3x + 2x \cdot (-2) + 3 \cdot 3x + 3 \cdot (-2)
\][/tex]
Simplifying this, we get:
[tex]\[
6x^2 - 4x + 9x - 6 = 6x^2 + 5x - 6
\][/tex]
2. Rewrite the Integral: Substitute the expanded form back into the integral.
[tex]\[
\int \frac{6x^2 + 5x - 6}{x^2} \, dx
\][/tex]
3. Simplify the Expression: Break down the fraction by dividing each term in the numerator by [tex]\(x^2\)[/tex]:
[tex]\[
\int \left( \frac{6x^2}{x^2} + \frac{5x}{x^2} - \frac{6}{x^2} \right) \, dx = \int \left( 6 + \frac{5}{x} - \frac{6}{x^2} \right) \, dx
\][/tex]
4. Integrate Term-by-Term: Integrate each term separately.
[tex]\[
\int 6 \, dx + \int \frac{5}{x} \, dx - \int \frac{6}{x^2} \, dx
\][/tex]
- The integral of [tex]\(6\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
6x
\][/tex]
- The integral of [tex]\(\frac{5}{x}\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
5\log|x|
\][/tex]
- The integral of [tex]\(\frac{-6}{x^2}\)[/tex] can be rewritten as [tex]\(-6x^{-2}\)[/tex]. The integral of [tex]\(x^n\)[/tex] is [tex]\(\frac{x^{n+1}}{n+1}\)[/tex]:
[tex]\[
\int -6x^{-2} \, dx = -6 \int x^{-2} \, dx = -6 \left( \frac{x^{-1}}{-1} \right) = \frac{6}{x}
\][/tex]
5. Combine the Results: Put all the integrated terms together.
[tex]\[
6x + 5\log|x| + \frac{6}{x}
\][/tex]
Therefore, the answer to the integral [tex]\(\int \frac{(2x + 3)(3x - 2)}{x^2} \, dx\)[/tex] is:
[tex]\[
6x + 5\log|x| + \frac{6}{x} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
