1 Example 1 Evaluate Solution Since the degree 2 of the numerator equals the degree of the denominator, we must begin with a long division: Thus Observe that the denominator of the integrand factors: x 2 +3x-28 = (x+7)(x-4). We will use the method of partial fractions to write Begin by adding the fractions on the right and equating the numerators:

2 We illustrate how to use each of the two methods of partial fractions to find the values of the numbers A and B. Method 1 Multiply out the right hand side of the above equation and collect the coefficients of each power of x: Equate the coefficients of x and the constant terms in the preceding equation to obtain the following linear system of 2 equations in 2 unknowns: From the first equation: B = -3-A. Substitute this value of B into the second equation: 28 = -4A + 7(-3-A) = -11A – 21. Hence 11A = -49 and A = -49/11. Then B= -3 – A = /11 = 16/11.

3 Method 2 Substitute x=-7 into the above equation: -3(-7) + 28 = A(-7-4) + 0, so 49 = -11A and A = -49/11. Substitute x=4 into the above equation: -3(4) + 28 = 0 + B(4+7), so 16 = 11B and B = 16/11. Clearly Method 2 is much easier to use than Method 1. By either of these methods: