Find or evaluate the integral by completing the square

Recall  that :

as the integrand function

as the antiderivative of

"a" as the lower boundary value of x

"b" as the upper boundary value of x

To evaluate the given problem: , we need to determine the

 indefinite integral F(x)  of the integrand: .

We apply completing the square on .

Factor out  from to get

The or resembles where:

and that we can plug-into .

              

             

             

To complete the square, we add and subtract 4 inside the ():

Distribute (-1) in "-4" to move it outside the ().

                         

Apply factoring for the perfect square trinomial: 

                                     

which means 

Applying it to the integral:

To solve for the indefinite integral of ,

let then and .

Apply u-substitution , we get:

                             

                             

 Apply the basic integration property: .  

 

For the integration of the first term:  ,

let then or then it becomes:

Applying radical property: and  Law of exponent: , we get:

Then,

Applying Power Rule of integration:

                         

                       

                         

                         

Recall .

Then,

For the integration of the second term:   ,

 we apply the basic integration formula for inverse sine function:

Then,

                    

 For the complete indefinite integral, we combine the results as:

 Then plug-in to express it terms of x, to solve for .

For the definite integral, we applying the boundary values: and in .