Sure, let's find [tex]\( f(x) \)[/tex] given the derivative [tex]\( f'(x) = 2x^3 - \varphi \)[/tex].
### Step-by-Step Solution:
1. Understand the Problem: We need to find [tex]\( f(x) \)[/tex] given its derivative. The derivative [tex]\( f'(x) \)[/tex] tells us how the function [tex]\( f(x) \)[/tex] changes with respect to [tex]\( x \)[/tex].
2. Given Derivative Expression:
[tex]\[
f'(x) = 2x^3 - \varphi
\][/tex]
3. Integrate the Derivative: To find [tex]\( f(x) \)[/tex], we need to find the antiderivative of [tex]\( f'(x) \)[/tex]. This means integrating the expression [tex]\( 2x^3 - \varphi \)[/tex] with respect to [tex]\( x \)[/tex].
4. Set Up the Integral:
[tex]\[
f(x) = \int (2x^3 - \varphi) \, dx
\][/tex]
5. Compute the Integral:
- Integral of [tex]\( 2x^3 \)[/tex]:
[tex]\[
\int 2x^3 \, dx = \frac{2}{4} x^4 = \frac{1}{2} x^4
\][/tex]
- Integral of [tex]\( \varphi \)[/tex]:
[tex]\[
\int -\varphi \, dx = -\varphi x
\][/tex]
6. Combine the Results:
[tex]\[
f(x) = \frac{1}{2} x^4 - \varphi x + C
\][/tex]
Here, [tex]\( C \)[/tex] is the constant of integration, which represents any constant value that could be added to [tex]\( f(x) \)[/tex] since the derivative of a constant is zero.
### Final Function:
Combining all parts together, we get:
[tex]\[
f(x) = \frac{1}{2} x^4 - \varphi x + C
\][/tex]
In this function, [tex]\( \varphi \)[/tex] is treated as a constant with respect to [tex]\( x \)[/tex].
