Integration by Partial Fractions

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Before using this technique of integration, let us recall what we have learnt about partial fraction.

Rational function

If P(x) and Q(x) are two polynomials in x, then the ratio of two polynomials,

is called a rational function, where

Proper rational function

If the degree of the numerator of the rational function is less than that of the denominator, the rational function is called a proper rational function.

Example is a proper rational function.

Improper rational function

If the degree of the numerator is greater than the degree of the denominator in a rational fraction, then the rational function is called improper rational function.

Like the case of improper fractions reducible to an integer added to a proper fraction, improper rational function can be reduced as a sum of a polynomial and a proper rational function.

In other words if is improper rational function, then

Where T(x) is a polynomial and is proper rational function.

Partial fractions

Any proper rational function can be expressed as sum of rational fractions, each having a factor of Q(x). Each such fraction is known as Partial fraction.

Example:

is a proper rational function.

This can be expressed as

are the partial fractions.

\ The given proper rational function is resolved into two simpler rational fractions.

Note that the denominators of the rational functions are resolved into factors.

Working rule for integration by parts

(1) Let be rational function. If is improper, divide P(x) by Q(x). Let T(x) be the quotient and P₁(x) be the remainder, then

Where T(x) is a polynomial and is a proper rational function.

(2) Resolve the proper rational function in to partial fractions.

(3) Write as sum of partial fractions.

(4) Write as the sum of T(x) and the sum of partial fractions. Integrate each part of the right hand side. This gives the required integral.

Note that if is a proper rational fraction,

Step 1 need not be performed.

The following table indicates the simpler partial fractions associated to proper rational functions.

In the above table A, B, C and D are real numbers to be determined suitably.

Example:

Integrate the following rational fraction .

Divide the numerator by the denominator, since the rational fraction is improper.

Resolve into partial fraction as follows

Put x =1, A = 3/2

Put x = -1, B = 1/2

Substitution the values of A and B in (2) we have

From (1), we have