TOPIC TECHNIQUES OF INTEGRATION

1. Integration by parts 2. Integration by trigonometric substitution 3. Integration by miscellaneous substitution 4. Integration by partial fraction

TECHNIQUES OF INTEGRATION 4. Integration by partial fraction

A rational function is a function which can be expressed as the quotient of two polynomial functions. That is, a function H is a rational function if where both f(x) and g(x) are polynomials. In general, we shall be concerned in integrating expressions of the form: DEFINITION

If the degree of f(x) is less than the degree of g(x), their quotient is called proper fraction; otherwise, it is called improper fraction. An improper rational function can be expressed as the sum of a polynomial and a proper rational function. Thus, given a proper rational function: Every proper rational function can be expressed as the sum of simpler fractions (partial fractions) which may have a denominator which is of linear or quadratic form.

The method of partial fractions is an algebraic procedure of expressing a given rational function as a sum of simpler fractions which is called the partial fraction decomposition of the original rational function. The rational function must be in its proper fraction form to use the partial fraction method. Four cases shall be considered. Case 1. Distinct linear factor of the denominator Case 2. Repeated linear factor of the denominator Case 3. Distinct quadratic factor of the denominator Case 4. Repeated quadratic factor of the denominator

For each linear factor of the denominator, there corresponds a partial fraction having that factor as the denominator and a constant numerator. Case 1. Distinct linear factor of the denominator That is, where A, B, …..N are constants to be determined Thus,

EXAMPLE: Evaluate each integral.

Case 2. Repeated linear factor of the denominator If the linear factor appears as the denominator of the rational function for each repeated linear factor of the denominator, there corresponds a series of partial fractions, where A, B, C, …, N are constants to be determined. The degree n of the repeated linear factor gives the number of partial fractions in a series. Thus,

EXAMPLE: Evaluate each integral.

For each non-repeated irreducible quadratic factor of the denominator there corresponds a partial fraction of the form. Case 3. Non-r epeated quadratic factor of the denominator where A, B, …..N are constants to be determined Thus,

EXAMPLE: Evaluate each integral.

For each repeated irreducible quadratic factor of the denominator there corresponds a partial fraction of the form. Case 4. Repeated quadratic factor of the denominator where A, B, …..N are constants to be determined Thus,

EXAMPLE: Evaluate each integral.

Evaluate each integral.