In calculus, an integral is a mathematical object that corresponds to summing infinitesimal data that may describe concepts such as displacement, area, and volume. The process of computing an integral is referred to as integration, and it is the inverse operation of differentiation.
Given a function f(x) that is continuous over the interval [a, b], the integral of the function over the interval represents the area under the graph of f(x), and is denoted as follows:
The a and b on the integral symbol denote the bounds of integration; f(x) is referred to as the integrand; dx is the variable of integration. The above can be read as "the integral from a to b of f(x)" where x is the variable of integration.
Integrals are described as either definite or indefinite integrals. The integral above is a definite integral because the limits of integration are defined. On the other hand, an indefinite integral has no such bounds. Rather, an indefinite integral is an anti derivative, or a function whose derivative is the function. An indefinite integral is denoted in much the same way as a definite integral, except that there are no bounds on the integral symbol:
The result of an indefinite integral is the anti derivative F(x) of the function f(x), plus some constant C:
The C term is necessary because the derivative of a constant is 0, so there are any number of anti derivatives that can satisfy the above equation. For example, given that f(x) = 3x2, one anti derivative for f(x) is x3, since, using the power rule, the derivative of x3 is 3x2. However, the derivative of x3 + 1, x3 - 4, x3 + ⅓, or x3 + C, are all also 3x2. Thus, finding the indefinite integral of a function means finding its set of anti derivatives. This is in contrast to a definite integral in which the result of integration is some number.
There are many well-defined rules for integrating commonly used functions. The table below shows some of the most common but does not include some of the more complex rules, such as integration by substitution or integration by parts.
|Multiplication by a constant|
|Power rule (n ≠ -1)|
Below are a few examples of using the integral rules to find the indefinite integral of a given function.
Evaluate the following expressions.
The square root term can be rewritten in the form of a power so that the power rule can be used, allowing the expression to be integrated as follows:
First, simplify the integrand:
The expression can then be integrated using the power rule: