Rules of integration, Mathematics

Rules of Integration

1. If 'k' is a constant then

∫Kdx

=  kx + c

2. In the above rule, if k = 1 then

∫dx  (this means integral of 1 which is written as dx and not 1 dx)

         = x + c

3. 

∫xndx = 52_rules of integration.png + c

*

- 1

The integral of 1/x or x-1 is

∫x1. dx =  ln x + c  x > 0

 

         The condition x > 0 is added because only positive numbers have logarithms.

4. 

∫akxdx =   1403_rules of integration1.png

+ c where 'a' and 'k' are constants.

 

5. 

∫eKxdx = 2443_rules of integration2.png + c since ln e = 1

2317_rules of integration3.png

Functions which differ from each other only by a constant have the same derivative. For example, the function F(x) = 4x + k has the same derivative, F'(x)= 4 = f(x), say, for any infinite number of possible values for k. If the process is reversed, it is clear that  ∫4dx is the indefinite integral for an infinite number of functions differing from each other only by a constant. The constant of integration, mentioned 'c' in the expression for integration earlier, thus represents the value of any constant which was part of the original function but precluded from the derivative by the rules of differentiation.

The graph of an indefinite integral ∫f(x)dx = F(x) + c, where 'c' is unspecified, is a family of curves parallel in the sense that the slope of the tangent to any of them at x is f(x). Specifying 'c' gives a single curve whereas changing 'c' shifts the curve vertically. If c = 0, the curve begins at the origin.

For example,  ∫4d(x)  = 4x + c. For c = -7, -3, 0, 1 and 4 the graph of this integral is given below.

Figure 

1868_rules of integration4.png
Posted Date: 9/13/2012 7:44:54 AM | Location : United States







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