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| Integration by Partial Fractions |
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| Before using this technique of integration, let us recall what we have learnt about partial fraction. |
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| If P(x) and Q(x) are two polynomials in x, then the ratio of two polynomials, |
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 is called a rational function, where  |
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| 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. |
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Example  is a proper rational function. |
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| 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. |
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| 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. |
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In other words if  is improper rational function, then |
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Where T(x) is a polynomial and  is proper rational function. |
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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. |
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| Example: |
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 is a proper rational function. |
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| This can be expressed as |
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 are the partial fractions. |
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| \ The given proper rational function is resolved into two simpler rational fractions. |
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| Note that the denominators of the rational functions are resolved into factors. |
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(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 |
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Where T(x) is a polynomial and  is a proper rational function. |
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(2) Resolve the proper rational function  in to partial fractions. |
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(3) Write  as sum of partial fractions. |
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(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. |
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Note that if  is a proper rational fraction, |
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| Step 1 need not be performed. |
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| The following table indicates the simpler partial fractions associated to proper rational functions. |
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| In the above table A, B, C and D are real numbers to be determined suitably. |
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| Example: |
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Integrate the following rational fraction  . |
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| Divide the numerator by the denominator, since the rational fraction is improper. |
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Resolve  into partial fraction as follows |
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| Put x =1, A = 3/2 |
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| Put x = -1, B = 1/2 |
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| Substitution the values of A and B in (2) we have |
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 From (1), we have |
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