To calculate the indefinite integral of [tex]\((2x + 3)^{3/5}\)[/tex] with respect to [tex]\(x\)[/tex], we can use a substitution to simplify the integration process. Here are the steps:
1. Substitute: Let [tex]\(u = 2x + 3\)[/tex]. This substitution changes the variable of integration, making it easier to integrate the function. Then, we differentiate [tex]\(u\)[/tex] with respect to [tex]\(x\)[/tex]:
[tex]\[
du = 2 \, dx \quad \Rightarrow \quad \frac{du}{2} = dx
\][/tex]
2. Rewriting the Integral: Replace the variables in the original integral:
[tex]\[
\int (2x + 3)^{3/5} \, dx = \int u^{3/5} \cdot \frac{du}{2}
\][/tex]
3. Factor Out the Constant: Pull the constant [tex]\(\frac{1}{2}\)[/tex] out of the integral:
[tex]\[
= \frac{1}{2} \int u^{3/5} \, du
\][/tex]
4. Integrate: Use the power rule for integration, which states [tex]\(\int u^n \, du = \frac{u^{n+1}}{n+1}\)[/tex], where [tex]\(n \neq -1\)[/tex]:
[tex]\[
\frac{1}{2} \int u^{3/5} \, du = \frac{1}{2} \cdot \frac{u^{(3/5)+1}}{(3/5) + 1}
\][/tex]
Simplify the exponent [tex]\((3/5) + 1\)[/tex]:
[tex]\[
(3/5) + 1 = (3/5) + (5/5) = \frac{8}{5}
\][/tex]
Therefore:
[tex]\[
\frac{1}{2} \cdot \frac{u^{8/5}}{8/5} = \frac{1}{2} \cdot \frac{5}{8} u^{8/5} = \frac{5}{16} u^{8/5}
\][/tex]
5. Substitute Back: Replace [tex]\(u\)[/tex] with [tex]\(2x + 3\)[/tex]:
[tex]\[
= \frac{5}{16} (2x + 3)^{8/5}
\][/tex]
Recall from our original context, to get our particular antiderivative result, we should confirm:
[tex]\[
\frac{5}{16}(2x+3)^{8/5} = 0.3125 (2x + 3)^{1.6}
\][/tex]
6. Result: Therefore, the indefinite integral is:
[tex]\[
\int (2x + 3)^{3/5} \, dx = 0.3125 (2x + 3)^{1.6} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
