Integration is an important concept in mathematics and—together with its inverse, differentiation—is one of the two main operations in calculus. Given a function $f$ of a real variable $x$, and an interval $[a, b]$ of the real line, the definite integral $\int_a^b \! f(x)\,dx$ is defined informally to be the area of the region in the $xy$-plane bounded by the graph of $f$, the $x$-axis, and the vertical lines $x=a$ and $x=b$, such that area above the $x$-axis adds to the total, and that below the $x$-axis subtracts from the total. The term integral may also refer to the notion of the anti-derivative, a function $F$ whose derivative is the given function $f$.

Definite Integral

A definite integral of a function can be represented as the signed area of the region bounded by its graph.

More rigorously, once an anti-derivative $F$ of $f$ is known for a continuous real-valued function $f$ defined on a closed interval $[a, b]$, the definite integral of $f$ over that interval is given by

$\displaystyle{\int_a^b \! f(x)\,dx = F(b) - F(a)}$

If $F$ is one anti-derivative of $f$, then all other anti-derivatives will have the form $F(x) + C$ for some constant $C$. The collection of all anti-derivatives is called the indefinite integral of $f$ and is written as 

$\displaystyle{\int f\; \mathrm d x = F(x) + C}$

Integration proceeds by adding up an infinite number of infinitely small areas. This sum can be computed by using the anti-derivative.

Properties

Linearity

The integral of a linear combination is the linear combination of the integrals.

$\displaystyle{\int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx}$

Inequalities

If $f(x) \leq g(x)$ for each $x$ in $[a, b]$, then each of the upper and lower sums of $f$ is bounded above by the upper and lower sums, respectively, of $g$:

$\displaystyle{\int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx}$

Additivity

If $c$ is any element of $[a, b]$, then:

$\displaystyle{\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx}$

Reversing Limits of Integration

If $a > b$,

$\displaystyle{\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx}$

Integration by Substitution

By reversing the chain rule, we obtain the technique called integration by substitution. Given two functions $f(x)$ and $g(x)$, we can use the following identity:

$\displaystyle{\int [f'(g(x)) \cdot g'(x)]\; \mathrm d x = f(g(x)) + C}$

or written in terms of the "dummy variable" $u = g(x)$:

$\displaystyle{\int f'(u)\; \mathrm d u = f(u) + C}$

If we are going to use integration by substitution to calculate a definite integral, we must change the upper and lower bounds of integration accordingly.