How does the indefinite integral relate to the antiderivative?

The indefinite integral of $2x$ is $x^2+C$. In this context, the term 'indefinite integral' is simply another name for the antiderivative, representing the family of all functions whose derivative is $2x$.

Conditions: The expression being integrated is $2x$.; The result includes an arbitrary constant $C$.

Integral notation is introduced early as a shorthand for antiderivatives to avoid writing cumbersome sentences. Its deeper geometric meaning, relating to areas under curves and sums of rectangles, will be motivated later when the course covers definite integrals.

Conditions: The current focus is on understanding antiderivatives.; The course will later cover definite integrals and areas under curves.

The notation $\int 2x \, dx$ is a compact way to ask for the antiderivative of $2x$ with respect to $x$. It represents the entire family of functions whose derivative is $2x$, which is written as $x^2+C$.

Conditions: The variable of integration is $x$.; The integrand is $2x$.

The general form of the antiderivative of $2x$ is $x^2+C$, where $C$ represents an arbitrary constant. This is because multiple functions differing only by a constant share the same derivative.

Conditions: The function being integrated is $2x$.; $C$ is an arbitrary constant.

Adding a constant to a function does not change its derivative because the derivative of a constant is zero. When differentiating a sum, the constant term vanishes, leaving the derivative of the original function unchanged.

Conditions: The function is differentiable.; The added term is a constant (e.g., 1, $\pi$, or $C$).