Some Integration Rules

Some Integration Rules

word/media/image1.gif

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.

word/media/image2.gif word/media/image3.gif (du + dv) = word/media/image3.gif du + word/media/image3.gif 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:

word/media/image3.gif (4x + 5x2) dx = word/media/image3.gif4x dx + word/media/image3.gif 5x2 dx

word/media/image2.gif word/media/image3.gif a du = a word/media/image3.gif 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 commutativeoperations. It should be noted that a variable cannot be transferred in this way. Thus,

word/media/image3.gif x dx is not equal to x word/media/image3.gif dx. 

Example:

word/media/image3.gif 3 x3 dx = 3 word/media/image3.gif x3 dx

word/media/image2.gif word/media/image3.gif un du = (un+1) / (n+1).

Example:

word/media/image3.gif x4 dx = x(4+1)/(4+1) = x5/5

Also, any one of these formulas can be proved by showing that the differential of the right member is equal to the expression under the integral sign.

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

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