Integration by Partial Fractions[br]Partial fraction decomposition can help you with differential equations of the following form:[br][br][img]https://quickmath.com/images/artimages/partia/partia1.jpg[/img][br][br]In solving this equation, we obtain[br][br][img]https://quickmath.com/images/artimages/partia/partia2.jpg[/img][br][br]The problem is that we have no technique for evaluating the integral on the left side.A technique called integration by partial fractions, in its broadest applications handles a variety of integrals of the form[br][br][img]https://quickmath.com/images/artimages/partia/partia3.jpg[/img][br][br]where p and q are polynomial functions. The technique of partial fractions becomes more complicated as the polynomials become more complicated. We shall illustrate the technique via some examples of special cases.[br][br][b]Example:[/b][br][img]https://quickmath.com/images/artimages/partia/partia4.jpg[/img][br][br]Solution Note that the denominator of the integrand can be factored:[br][br][img]https://quickmath.com/images/artimages/partia/partia5.jpg[/img][br][br]The plan is to decompose this fraction into partial fractions by finding numbers A and B for which[br][br][img]https://quickmath.com/images/artimages/partia/partia6.jpg[/img][br][br]holds for all x except x = 1 and x = - 2. If this is possible, then we can integrate 1/(x^2+x-2) by finding :[img]https://quickmath.com/images/artimages/partia/partia7.jpg[/img][br][br]since these last two antiderivatives can be evaluated easily in terms of the natural logarithm.[br]We shall now show how to find A and B. Note that if we multiply both sides of the equation[img]https://quickmath.com/images/artimages/partia/partia8.jpg[/img][br][br][br]by (x - l)*(x + 2), we obtain1 = A (x + 2) + B (x - 1).The last equation must hold for all x, that is, it is an identity. Since it holds for all x, it must hold for any specific values of x that we choose. Observe that if we choose x = - 2, then the term involving A will become 0, and we havel = A(-2+2)+B(-2-1)= -3Bfrom which we immediately get B = -1/3 . If we next choose x = 1, we have1 = A (1+2)+B(1-1) = 3A,and consequently A = 1/3 . Substituting these values of A and B into Formula (2), we obtain[br][br][img]https://quickmath.com/images/artimages/partia/partia9.jpg[/img][br][br][br]Thus, we use partial fractions to express the fraction on the left in Equation (2).[br]We can now complete the integration problem.[br][br][br][img]https://quickmath.com/images/artimages/partia/partia10.jpg[/img]