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







Related Discussions:- Rules of integration, Assignment Help, Ask Question on Rules of integration, Get Answer, Expert's Help, Rules of integration Discussions

Write discussion on Rules of integration
Your posts are moderated
Related Questions
On a picnic outing, 2 two-person teams are playing hide-and-seek. There are four hiding locations (A, B, C, and D), and the two peoples of the hiding team can hideseparately in any

A round balloon of radius 'a' subtends an angle θ at the eye of the observer while the angle of elevation of its centre is Φ.Prove that the height of the center of the balloon is a

Children Learn By Experiencing Things : One view about learning says that children construct knowledge by acting upon things. They pick up things, throw them, break them, join the

Peggy's town has an average temperature of 23° Fahrenheit in the winter. What is the average temperature on the Celsius scale? If the total amount for both is 80, after that th

if 1/x+2, 1/x+3, 1/x+5 are in AP find x Ans 1/x+2,1/x+3, 1/x+5 are in AP find x. 1/x+3 - 1/x+2 = 1/x+5-1/x+3 =>  1/x 2 +5x+6  = 2/ x 2 +8x +15 =>  On solving we get x

Q. How to Subtract fractions with different denominators? Ans. As with adding fractions, you can't subtract unless the denominators are the same. Here is an example: 9/

Differentiate following functions.                       h (t ) = 2t 5 + t 2 - 5 / t 2 We can simplify this rational expression as follows.                       h (t )


Computation of Covariance Ungrouped Data          For a population consisting of paired ungrouped data points {X, Y} where,

i have trouble going through problem in this lesson. Markdown and Markups are theh ones im stuck in