Rules of Integration for Indefinite Integral

Method of Partial Fractions for Indefinite Integral

where Φ(x) and φ(x) are rational, integral, algebraical functions of x

The integrand after denominator factorization is

Integration by Partial Fractions

The method of integration by partial fractions can be expressed as

The integral of polynomial Q(x) can be obtained by making use of the constant multiple, sum of function properties of intergration and applying the anti-differentiation of the derivative of polynomial. The integration of the four case of fraction factors after partial fractions are:

1. linear partial fraction A/(ax+b)

   This is the most common type of partial fraction, the integral of the linear partial fraction can be obtained by the quotient of standard functions rule of indefinite integral,

   

   Therefore the integral of linear partial fraction can be determined by:

   

2. repeated linear partial fraction  A/(ax+b)k

   This is the most simple type of partial fraction.The integral of the repeated linear partial fraction can be obtained by product of standard functions rule of indefinite integral.

   

   Therefore the integral of repeated linear partial fraction can be determined by:

   

3. quadratic partial fractionr (Ax+B)/(ax2+bx+c)

   For the quadratic partial fraction, there are some variant forms.

   1. Standard quadratic partial fraction (Ax+B)/(ax2+bx+c).

      1. If the numerator can be expressed as the derivative of the denumerator, imply

         

      2. If the numerator can not be expressed as the derivative of the denumerator, imply

         

         For the last integral, the quadratic factor can be resolved by completing the square, imply

         

         Since the quadratic factor is irreducible, 4ac-b2>0, imply

         

         Therefore the integral is

         

   2. Quadratic partial fraction Ax/(ax2+bx+c). Same as a.ii case where B=0, imply

      

   3. Quadratic partial fraction B/(ax2+bx+c). Same as the last integral in a.ii case where B=D, imply

      
   4. Quadratic partial fraction with quadratic factor, (ax2+bx)

      Quadratic factor (ax2+bx) is same as (ax2+bx+c) by letting c=0, the integral can be obtained as in case a

   5. Quadratic partial fraction with quadratic factor, (ax2+c)

      Quadratic factor (ax2+c) is same as (ax2+bx+c) by letting b=0, the integral can be obtained as in case a. But no complete the square is needed since the factor can be directly transformed into needed format., imply

      

4. repeated quadratic factor  (Ax+B)/(ax2+bx+c)k

   The proceduce of determine the integral is same as case 3 except for the determining of the integral with power after decomposing the original integral into two integrals. Imply

   

   The first integral can be obtained by simple substitution. Imply

   

   The second integral can be obtained by simple substitution. Imply

   

   The second integral can further be expanded using the formule in integration by part as:

   

   The theta angle can be transformed back to x through trigonometry, imply:

   

   And the integral can be expressed as: