PURE MATHEMATICS - Integration

 

Integration "By Parts"

 

 

'By Parts' Method

example #1

example #2

example #3

 

 

 

Integration 'By Parts' - from the Product Rule

 

The integration of expressions where there are two separate functions multiplied together, is essentially by an amended version of Leibnitz's Product Rule.

 

To integrate an expression of the type:

 

       

 

make       and   

 

the expression becomes:      

 

using the Product Rule:       

 

    

 

                       rearranging         

 

integrating with respect to x

 

      

 

 

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Example #1

integrate      

 

make      

 

then     

 

    so        

 

 

make     

 

         so    

    hence     

 

 

now using the formula   

 

 

 

       

 

         

 

 

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Example #2

integrate    

 

make      

 

then   

 

             

 

 

  make    

 

           then       

 

     integrating                    

 

 

now using the formula   

 

             

                       

                 

 

 

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Example #3

integrate    

 

 

make    

 

then         

 

    

 

 

 make  

 

      then   

 

  integrating           

 

now using the formula   

 

 

 

                   

 

               

 

                    

 

 

 

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