To solve the integral [tex]\(\int (x^{3/2} + 2x + 1) \, dx\)[/tex], let's address it term by term.
1. Integrate [tex]\(x^{3/2}\)[/tex]:
- Recall the power rule for integration: [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex], where [tex]\(C\)[/tex] is a constant.
- For [tex]\(x^{3/2}\)[/tex], [tex]\(n = \frac{3}{2}\)[/tex].
- Applying the power rule:
[tex]\[
\int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} = \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2}
\][/tex]
2. Integrate [tex]\(2x\)[/tex]:
- Again, using the power rule where [tex]\(n = 1\)[/tex]:
[tex]\[
\int 2x \, dx = 2 \int x \, dx = 2 \left( \frac{x^2}{2} \right) = x^2
\][/tex]
3. Integrate 1:
- The integral of a constant [tex]\(a\)[/tex] is simply [tex]\(ax + C\)[/tex]:
[tex]\[
\int 1 \, dx = x
\][/tex]
Putting all parts together, summing the antiderivatives:
[tex]\[
\int (x^{3/2} + 2x + 1) \, dx = \frac{2}{5} x^{5/2} + x^2 + x + C
\][/tex]
Where [tex]\(C\)[/tex] is the constant of integration.
Therefore, the integral of the given function [tex]\(x^{3/2} + 2x + 1\)[/tex] is:
[tex]\[
\int (x^{3/2} + 2x + 1) \, dx = x^2 + x + \frac{2}{5} x^{5/2} + C
\][/tex]
This confirms that the integral of the function yields [tex]\(x^2 + x + 0.4 x^{2.5}\)[/tex].
