Page:Calculus Made Easy.pdf/141



Partial Fractions.

have seen that when we differentiate a fraction we have to perform a rather complicated operation; and, if the fraction is not itself a simple one, the result is bound to be a complicated expression. If we could split the fraction into two or more simpler fractions such that their sum is equivalent to the original fraction, we could then proceed by differentiating each of these simpler expressions. And the result of differentiating would be the sum of two (or more) differentials, each one of which is relatively simple; while the final expression, though of course it will be the same as that which could be obtained without resorting to this dodge, is thus obtained with much less effort and appears in a simplified form.

Let us see how to reach this result. Try first the job of adding two fractions together to form a resultant fraction. Take, for example, the two fractions $$\dfrac {1}{x+1}$$ and $$\dfrac {2}{x-1}$$. Every schoolboy can add these together and find their sum to be $$\dfrac {3x+1}{x^2-1}$$. And in the same