Integral calculus is mainly concerned with summing

the values of a function over a particular range,

and is particularly  useful for finding the area of an irregular shape.

 

·       First order Differential Equations

 

 

 

 

 

 

Example

 

Find the particular solution of the differential equation

 

     given by y = 5 when x = 3

 

 

 

 

2x²-2x-7 is called the anti-derivative of  4x-2

since     

 

 

·       Integration

 

Integration is the process of finding the anti-derivative.

 

        

 

  means the indefinite integral of the function

is to be calculated with respect to x.

The anti-derivative F(x) is called the integral,

c is called the constant of integration.

 

 

 so  differentiating the integral results

in the original function.

 

 

·       Basic Rules of Integration

 

   

 

 

Examples

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

·       ∫(ax+b)ⁿ

 

 

 

 

Examples

 

 

 

 

 

 

 

 

 

·       Trig functions

 

Remember to use   Radians !!!

 

 

Examples

 

 

 

   

 

 

 

 

 

 

 

·       Logarithms and Exponentials

 

              

 

Don’t forget to apply the chain rule

in reverse if necessary!

 

 

Examples

 

 

 

 

 

     

 

 

 

 

 

 

 

 

·       Integration by substitution

 

 

   

Examples

 

  

 

  

 

 

 

 

 

 

 

 

 

 

 

 

Sometimes it is easier to substitute dx

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Common Forms

 

 

 

Examples

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Examples

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Examples