13Partial FractionsThe method of Partial Fractions provides a way to integrate all rational functions. Recall that a rational function is a function of the formwhere P and Q are polynomials.The technique requires that the degree of the numerator (pembilang) be less than the degree of the denominator (penyebut)If this is not the case then we first must divide the numerator into the denominator.
142. We factor the denominator Q into powers of distinct linear terms and powers of distinct quadratic polynomials which do not have real roots.3. If r is a real root of order k of Q, then the partial fraction expansion of P/Q contains a term of the formwhere A1, A2, ..., Ak are unknown constants.
154. If Q has a quadratic factor ax2 + bx + c which corresponds to a complex root of order k, then the partial fraction expansion of P/Q contains a term of the formwhere B1, B2, ..., Bk and C1, C2, ..., Ck are unknown constants.5. After determining the partial fraction expansion of P/Q, we set P/Q equal to the sum of the terms of the partial fraction expansion. (See Ex-2.Int.Frac)
166. We then multiply both sides by Q to get some expression which is equal to P. 7. Now, we use the property that two polynomials are equal if and only if the corresponding coefficients are equal.(see ex3-int.Fractional)8. We express the integral of P/Q as the sum of the integrals of the terms of the partial fraction expansion.(see Ex4-Int.Fractional)