Thursday, December 1, 2011

Find or evaluate the integral by completing the square

 To evaluate the given integral:  ,


 we follow the first fundamental theorem of calculus: 


If f is continuous on closed interval [a,b], we follow:



 where F is the anti-derivative or indefinite integral of f on closed interval .


  To determine the , we apply completing the square on the trinomial:


Completing the square:


is in a form of  


 where:


a =1


b =4


 c= 13


 To complete square ,we add and subtract on both sides:


With a=1 and b = 4 then:



Then becomes:




Applying in the given integral, we get:



 The integral form: resembles the 


basic integration formula for inverse tangent function:



Using u-substitution, we let then   or  


where the boundary  limits:  upper bound = 2 and lower bound =-2


and   then


The indefinite integral will be:



           


Plug-in to solve for :




We now have  



Applying  , we get:



 




  

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