Rules of integration, Mathematics

Assignment Help:

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

Related Discussions:- Rules of integration

Matrices, (e) Solve the following system of equations by using Matrix meth...

(e) Solve the following system of equations by using Matrix method. 3x + 2y + 2z = 11 x + 4y + 4z = 17 6x + 2y + 6z = 22

In an election contested between a and b determine vote, In an election con...

In an election contested between A and B, A obtained votes equal to twice the no. of persons on the electoral roll who did not cast their votes & this later number was equal to twi

Determine the velocity and position functions of object, Determine if the a...

Determine if the acceleration of an object is given by a → = i → + 2 j → + 6tk → find out the object's velocity and position functions here given that the initial velocity is v

Laws of logarithms, express each logariths in terms of log3 P and log3 Q. 1...

express each logariths in terms of log3 P and log3 Q. 1. log3 P^2 Q^3

INVESTING MONEY, HOW MANY SHARES CAN I BUY WITH 1000 DOLLARS

HOW MANY SHARES CAN I BUY WITH 1000 DOLLARS

Find out how much acid solution mixed, A chemist has one solution which is ...

A chemist has one solution which is 50% acid and a second which is 25% acid. How much of each should be mixed to make 10 litres of 40% acid solution.

Continuous uniform distribution, Continuous Uniform Distribution Consid...

Continuous Uniform Distribution Consider the interest earned on a bank deposit. Let X equal the value after the decimal point. (Assume no rounding off to the nearest paise.) Fo

Find the probability, A bag contains 19 tickets, numbered from 1 to 19. A t...

A bag contains 19 tickets, numbered from 1 to 19. A ticket is drawn and then another ticket is drawn without replacement .Find the probability that both tickets will show even numb

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd