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:

...

 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: