1. 1 Example 3 Evaluate Solution Since the degree 5 of the numerator is greater than the degree 4 of the denominator, we begin with long division: Hence The denominator factors into linear factors: x 4 -13x 2 +36 = (x 2 -4)(x 2 -9) = (x-2)(x+2)(x-3)(x+3).

2. 2 We use the method of partial fractions to write the integrand Begin by adding the fractions on the right, and then equate the numerators of the resulting fraction and the fraction on the left: We use the second variation of the method of partial fractions. Let x=2 in the preceding equation: -60 = A(4)(-1)(5)+0+0+0= -20A and A = 3. Let x=-2 in this equation: -20 = 0+B(-4)(-5)(1)+0+0 = 20B and B = -1. Let x=3 in this equation: -120 = 0+0+C(1)(5)(6)+0 = 30C and C = -4. Let x=-3 in this equation: -60 = 0+0+0+D(-5)(-1)( -6) = -30D and D = 2. x 4 -13x 2 +36 = (x-2)(x+2)(x-3)(x+3)

3. 3 Thus A=3, B=-1, C=-4, D=2