6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

An antiderivative of a function f(x) is a function F(x) such that F'(x)=f(x). An indefinite integral represents the family of all antiderivatives of a function and is written with the symbol ∫. Because the derivative of any constant is 0, indefinite integrals must always include +C, called the constant of integration. So if one antiderivative of a function is x³, then the full family is x³+C. Unlike a definite integral, an indefinite integral has no bounds and gives a function family rather than a single numerical value.

In this lesson, you reverse derivative rules to find antiderivatives. The main rules to know are the constant rule, constant multiple rule, sum and difference rules, power rule, and the standard antiderivatives of e^x, 1/x, sin x, and cos x. These are the basic direct antiderivatives you use before substitution and more advanced techniques later. Here are the core rules:

\int k\,dx = kx+C

\int [cf(x)]\,dx = c\int f(x)\,dx

\int [f(x)+g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx

\int [f(x)-g(x)]\,dx = \int f(x)\,dx - \int g(x)\,dx

\int x^n\,dx = \frac{x^{n+1}}{n+1}+C \quad \text{for } n\ne -1

\int \frac{1}{x}\,dx = \ln|x|+C

\int e^x\,dx = e^x+C

\int a^x\,dx = \frac{a^x}{\ln a}+C \quad \text{for } a>0,\ a\ne1

\int \cos x\,dx = \sin x+C

\int \sin x\,dx = -\cos x+C

\int \sec^2 x\,dx = \tan x+C

\int \csc^2 x\,dx = -\cot x+C

\int \sec x\tan x\,dx = \sec x+C

\int \csc x\cot x\,dx = -\csc x+C

If you know these derivative-antiderivative pairs well, then most early integration problems become pattern recognition: you identify the form, reverse the derivative rule, and add +C.

In this unit

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