4.1 Introduction to Antiderivatives

The '$+C$' term represents the arbitrary constant of integration. It signifies that on a connected interval, all antiderivatives of a given function differ from one another by a constant.

Conditions: The domain is a connected interval.; The function being integrated has at least one antiderivative.

Different functions such as $x^2$ and $x^2 + C$ share the same derivative because the derivative measures the slope of the tangent line at each point $x$. Adding a constant $C$ shifts the graph vertically but does not change its steepness or shape.

Conditions: The functions differ only by an additive constant.; The domain is an interval where the derivative exists.

The general antiderivative of $\cos x$ is $\sin x + C$, and the general antiderivative of $e^x$ is $e^x + C$. In both cases, $C$ represents an arbitrary constant, indicating that any vertical shift of these base functions is also a valid antiderivative.

Conditions: The variable is $x$.; The domain is all real numbers (or a connected interval thereof).

In indefinite-integral notation, the integral sign ($\int$) indicates the operation of finding an antiderivative. The differential term ($dx$) identifies the variable of integration.

Conditions: Standard calculus notation is used.; The context is indefinite integration.

To verify an antiderivative, you differentiate the proposed result. If the derivative of the proposed function equals the original integrand, then the proposal is correct.

Conditions: The proposed function is differentiable.; The original integrand is defined on the domain of interest.