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Suppose you have a fraction whose rational form is x/y where y is a positive integer. Furthermore, suppose that y = p*q where p and q are also positive integers (factors of y). Then x/y, in partial fraction form is: a/p + b/q for some integers a and b.
This may seem rather a tedious sort of exercise, but it comes into its own when analysing algebraic fractions, particularly for integration.
For example, suppose you need to find the integral of (3x + 2)/(x^2 + 3x + 2). A linear expression divided by a quadratic is not readily integrated. But
(3x + 2)/(x^2 + 3x + 2) = (3x + 2)/[(x + 1)*(x + 2)] which, in terms of partial fractions, is
-1/(x + 1) + 4/(x + 2)
Integrating, I[(3x + 2)/(x^2 + 3x + 2)] = I[-1/(x + 1) + 4/(x + 2)] = I[-1/(x + 1)] + i[4/(x + 2)]
= -ln(x + 1) + 4*ln(x + 2) + c where ln is the natural logarithm and c is a constant of integration
= ln[(x+2)^4 /(x+1)] + c
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