To find the integral [tex]\(\int \frac{4x^5 + 3}{2x^2} \, dx\)[/tex], we can first simplify the integrand and then integrate the resulting expression step by step. Here is a detailed solution:
1. Simplify the integrand: We start by dividing each term in the numerator by the denominator:
[tex]\[
\frac{4x^5 + 3}{2x^2} = \frac{4x^5}{2x^2} + \frac{3}{2x^2}
\][/tex]
Simplifying each term:
[tex]\[
\frac{4x^5}{2x^2} = 2x^3
\][/tex]
[tex]\[
\frac{3}{2x^2} = \frac{3}{2} \cdot x^{-2}
\][/tex]
Therefore, the integrand simplifies to:
[tex]\[
2x^3 + \frac{3}{2}x^{-2}
\][/tex]
2. Set up the integral: We can now rewrite the integral with the simplified integrand:
[tex]\[
\int \left( 2x^3 + \frac{3}{2}x^{-2} \right) dx
\][/tex]
3. Integrate term by term:
- For the first term, [tex]\(2x^3\)[/tex]:
[tex]\[
\int 2x^3 \, dx
\][/tex]
Using the power rule of integration [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex]:
[tex]\[
\int 2x^3 \, dx = 2 \cdot \frac{x^{3+1}}{3+1} = 2 \cdot \frac{x^4}{4} = \frac{1}{2} x^4
\][/tex]
- For the second term, [tex]\(\frac{3}{2}x^{-2}\)[/tex]:
[tex]\[
\int \frac{3}{2}x^{-2} \, dx
\][/tex]
Again, using the power rule:
[tex]\[
\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} = \frac{x^{-1}}{-1} = -\frac{1}{x}
\][/tex]
Scaling by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
\int \frac{3}{2}x^{-2} \, dx = \frac{3}{2} \cdot \left( -\frac{1}{x} \right) = - \frac{3}{2x}
\][/tex]
4. Combine the results:
Putting it all together, the integral becomes:
[tex]\[
\int \left( 2x^3 + \frac{3}{2}x^{-2} \right) dx = \frac{1}{2}x^4 - \frac{3}{2x}
\][/tex]
Therefore, the result of the integral is:
[tex]\[
\int \frac{4x^5 + 3}{2x^2} \, dx = \frac{1}{2}x^4 - \frac{3}{2x} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
