Example of using Partial Fraction to evaluate Integral

 

Problem

 

          We like to evaluate the integral:

                 

  where n = 1, 2, 3, 4.

 

We employ the technique of Partial Fractions. This algebraic method is assumed to be known.

 

 

When n = 1,

 

         

 

When n = 2,

 

       By using the substitution  x = tan q, it is a simple exercise to get:

              

Then,

              

In getting (3) here, we change the variable and apply (2).

Alternatively, you can start by using the substitution  x = a tan q  at the beginning of (3).

 

 

When n = 3,

 

              

              

              

              

 

 

 

 

 

  A Partial Fraction exercise!

 

 

  Second fraction is further broken.

  Note : d(x2 – x + 1) = 2x – 1

 

  Use completing square in the denominator of the second fraction.

 

 

  In the second integral, you may use 2x – 1 = Ö3 tan q  to evaluate.

 

Finally you may check by changing the valuable suitably:

 

 

When n = 4,

 

Since  x4 + 1 cannot be factorized under rational numbers, we start with the factorization :

       x4 + 1 = (x4 + 2x2 + 1) – 2x2 = (x2 + 1)2 – (Ö2 x)2 = (x2 + Ö2 x + 1)(x2 -Ö2 x +1)

 

As a Partial Fraction exercise, we get:

 

      

              

              

              

      

Finally, if you still feel energetic of checking, we arrive: