Integration is usually performed by means of general formulae of which the following will be sufficient for basic applications. In these a, n, c are constants and u,v functions of a single variable.

(du + dv) = du + dv.

The integral of a sum is equal to the sum of the integrals of its parts. Integration, like differentiation, is thus a distributive operation.

Example:

(4x + 5x²) dx = 4x dx + 5x² dx

a du = a du.

A constant factor may be transferred from one side of the integral sign to the other without changing the result. Integration and multiplication by a constant are thus commutative operations. It should be noted that a variable cannot be transferred in this way. Thus, x dx is not equal to x dx.
Example:

3 x³ dx = 3 x³ dx

uⁿ du = (uⁿ⁺¹) / (n+1).

Example:

x⁴ dx = x⁽⁴⁺¹⁾/(4+1) = x⁵/5

By using the rules given on this page and the integrals of some common functions shown on the next page we will be able to compute the integrals of many elaborate functions.

Do you want to see the graphs of the integrals of a few functions?
Graphing interactive workbench