The intergral is the inverse operation of the derivative. Because of this, it is often called an antiderivative. We call this process integration because the result is an integrated sum of a series of infinitessimally thin slices of a curve.

Definite Integrals

With \(a\) and \(b\) numbers and \(n\) a positive integer, let:

\(Δ_nx=\frac{b-a}{n}\) and \(x_k=a+kΔ_nx\) for \(k=1,2,3,...,n\)

If \(f\) is a function that is continuous in the closed interval with endpoints \(a\) and \(b\), then both of the limits:

\(\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}f(x_k)Δ_nx\) and \(\lim\limits_{n\to\infty}\sum\limits_{k=0}^{n-1}f(x_k)Δ_nx\)

exist and have the same value. The value of these limits is called the definite integral of \(f\) from \(a\) to \(b\) and is represented by \(\int_{a}^{b}f(x)\,dx\), such that:

\(\int_{a}^{b}f(x)\,dx\) \(=\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}f(x_k)Δ_nx\) \(=\lim\limits_{n\to\infty}\sum\limits_{k=0}^{n-1}f(x_k)Δ_nx\)

The symbolic form \(f(x)\) is referred to as the integrand, \(a\) is called the lower limit of integration, and \(b\) is the upper limit of integration. The variable of integration may differ from one application to the next, and is specified as follows:

\(\int_{a}^{b}f(x)\,dx\), \(\int_{a}^{b}f(t)\,dt\), \(\int_{a}^{b}f(u)\,du\), etc.

The \(dx, dt, du,\) etc., specifies the variable of integration. This is important when more than one variable is present in an equation. For example, in the equation \(\int_{1}^{5}(x+y^2)\,dx\), \(x\) is the variable of integration, and not \(y\).

When performing an integration, it is possible to avoid introducing \(Δ_nx\) and \(x_k\), by writing the equation outright. For example:

\(\int_{a}^{b}f(x)\,dx\) \(=\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}f(a+k\frac{b-a}{n})\frac{b-a}{n}\) \(=\lim\limits_{n\to\infty}\sum\limits_{k=0}^{n-1}f(a+k\frac{b-a}{n})\frac{b-a}{n}\)

It should be noted that the lower limit of integration need not be less than the upper limit of integration, as swapping the two values simply changes the sign of the result; that is:

\(\int_{a}^{b}f(x)\,dx\) \(=-\int_{b}^{a}f(x)\,dx\)

Indefinite Integrals

The limit of a sum method of integration described above will always work; but as with the long-hand method of differentiation, there are quicker, easier methods involving antiderivatives or indefinite integrals.

The the differential of area method of integration, formally known as the fundamental theorem of calculus, states that if the integral \(\int f(x)\,dx\) exists, and a function \(F(x)\) also exists for which \(F^′(x)=f(x)\) in the interval \([a,b]\), then:

\(\int_{a}^{b}f(x)\,dx\) \(=F(b)-F(a)\)

Where the function \(F(x)\) is the indefinite integral of the function \(f(x)\).

Now, let \(f\) be a function and \(g\) be a primitive for \(f\). The symbol:

\(\int f(x)\,dx\)

denotes the general primitive of \(f\), namely \(g(x)\)+c, where \(g^′=f\); or put more clearly, \(g^′\) denotes that the function \(f\) is a derivative. We then call the general primitive the indefinite integral or antiderivative of \(f\). The definition of \(\int f(x)\,dx\) only refers to differentiation and not integration. The symbol was chosen to suggest integration because of the following connection between primatives and integrals:

\(\int f(x)\,dx\) \(=\int_{a}^{x} f(t)\,dt\), \(\int_{a}^{b} f(x)\,dx\) \(=\bigg[\int f(x)\,dx\bigg]_{a}^{b}\)

For example, the statement:

\(\int \frac{x}{\sqrt{1-x^2}}\,dx\) \(=-\sqrt{1-x^2}+c\)

says that the general primative or indefinite integral of the function \(\int \frac{x}{\sqrt{1-x^2}}\,dx\) is \(-\sqrt{1-x^2}+c\); or in other words, \(\frac{x}{\sqrt{1-x^2}}\) is the derivative of the primitive \(-\sqrt{1-x^2}+c\). This can be verified by differentiating the primitive.

Properties of Indefinite Integrals

In general, the following properties hold true:

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

\(cf(x)\,dx\) \(=c\int f(x)\,dx\)(\(c\) a constant)

\(x^n\,dx\) \(=\frac{x^{n+1}}{n+1}+c\)(\(n\) a number \(\ne 0\))

The value \(+c\) in the primitive is called the constant of integration and may be specified or omitted and implied; in either case, it always exists.

It should be noted that not all integrals can be evaluated by elementary methods. For example, \(\int e^{-x^2}\,dx\), \(\int sin\,x^2\,dx\), and \(\int\sqrt{1+x^3}\,dx\), have no known primitive and require special methods to integrate.

Typically, tables of predetermined integrals are used to speed up the process of integration whenever possible. Some tables are quite extensive and can be found at the library or online. A table of integrals can be found at the bottom of this page.


Table of Indefinite Integrals

Calculus: Derivatives