To find the function [tex]\( f(x) \)[/tex] given its derivative [tex]\( f'(x) = 3x^2 + 2x \)[/tex], we need to integrate the given derivative. Here's the step-by-step process:
1. Set up the integral: We need to find the antiderivative (indefinite integral) of [tex]\( f'(x) \)[/tex].
[tex]\[
f(x) = \int (3x^2 + 2x) \, dx
\][/tex]
2. Integrate each term separately: We break down the integral into parts and integrate each term individually.
[tex]\[
\int 3x^2 \, dx + \int 2x \, dx
\][/tex]
3. Integrate [tex]\( 3x^2 \)[/tex]: Using the power rule for integration, [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex], where [tex]\( C \)[/tex] is the constant of integration, we get:
[tex]\[
\int 3x^2 \, dx = 3 \left(\frac{x^{2+1}}{2+1}\right) = 3 \left(\frac{x^3}{3}\right) = x^3
\][/tex]
4. Integrate [tex]\( 2x \)[/tex]: Similarly, applying the power rule for the term [tex]\( 2x \)[/tex]:
[tex]\[
\int 2x \, dx = 2 \left(\frac{x^{1+1}}{1+1}\right) = 2 \left(\frac{x^2}{2}\right) = x^2
\][/tex]
5. Combine the integrated terms: Summing the results of the integrals, we get:
[tex]\[
f(x) = x^3 + x^2 + C
\][/tex]
Here, [tex]\( C \)[/tex] is the constant of integration.
Therefore, the resulting function [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = x^3 + x^2 + C
\][/tex]
In this context, without additional initial conditions or boundary conditions to determine the specific value of [tex]\( C \)[/tex], the general solution to the integrated function is:
[tex]\[
f(x) = x^3 + x^2
\][/tex]
