Antiderivatives and Indefinite Integrals

An antiderivative of a function f(x) is a function F(x) whose derivative is f(x). In other words, F'(x) = f(x). Finding an antiderivative is the reverse process of differentiation.

The indefinite integral of a function f(x) is the collection of all its antiderivatives. It is denoted by the integral symbol:

∫ f(x) dx = F(x) + C

where F(x) is *any* particular antiderivative of f(x), and C is the constant of integration. This constant represents the fact that the derivative of any constant is zero, so any function F(x) + C will have the same derivative f(x).

They are very closely related but not exactly the same:

• An **antiderivative** is a *single* function whose derivative is the original function.
• An **indefinite integral** represents the *set* of all possible antiderivatives of the original function, indicated by the addition of the constant of integration (+ C).

So, an indefinite integral is the *most general form* of an antiderivative. While the terms are often used interchangeably in informal contexts, technically the indefinite integral encompasses the entire family of antiderivatives.

The constant of integration, C, is important because the derivative of any constant is zero. This means if you have a function F(x), its derivative is F'(x) = f(x). If you have F(x) + 5, its derivative is also F'(x) + 0 = f(x). If you have F(x) - 10, its derivative is still f(x).

When finding an antiderivative or indefinite integral, you are reversing the differentiation process. Since the constant term disappeared during differentiation, you cannot uniquely determine it during integration unless you have extra information (like an initial condition or a point the function passes through). The + C represents this unknown constant and signifies the entire family of functions that have f(x) as their derivative.

Finding antiderivatives relies on knowing the rules of differentiation in reverse. For example:

• If the derivative of xⁿ⁺¹ is (n+1)xⁿ, then the antiderivative of xⁿ (for n ≠ -1) is xⁿ⁺¹ ÷ (n+1). (This is the Power Rule for integration).
• If the derivative of sin x is cos x, then the antiderivative of cos x is sin x.
• If the derivative of e^x is e^x, then the antiderivative of e^x is e^x.

You also use rules for sums, differences, and constant multiples, similar to differentiation. For more complex functions, techniques like substitution, integration by parts, or partial fraction decomposition are used.