If A. B. and C are II x 11 symmetric matrices. prol'e the following: (a) A is congruent to A. (b ) If B is congruent 10 A. then A is congruent to B. (e) If C is congruent to Band B is congruent to A, then C is congruent to A.

Chapter 8: Techniques of Integration Continued ____________________________________________________________ 8.4 Partial Fractions 1. The degree of f(x) is < g(x); otherwise use long division first. 2. g(x) can be factored How Partial Fractions Work 1. If(x− r) is a factor of g(x), we assign m partial fractions: m 2. If (x + px + q)is a factor of g(x), we assign m partial fractions: 3. Take the sum of the partial fractions ; let the sum be equal to f(x)/g(x) 4. In the numerator, equate the coefficients of x, then solve the system of equations for A, B, C, and so on Note: The power shows how many partial fractions to assign Example 1: Example 2: Example 3: Example 4: Short Cut For Partial Fractions 1. The degree of f(x) is < g(x) 2. g(x) can be factored 3. All factors are linear Example 1: Example 2: ____________________________________________________________ 8.7 Improper Integrals 1. Type I: Suppose you want to find the area under y = 1/x from x = 1 to ∞. Example 1: Example 2: Example 3: 2. Type II: - f(x) is not continuous at C where C is a number in [a,b] - f(x) has a vertical asymptote at C Example 1: Example 2: 3. Convergence: ∞ If ∫ is finite, the integral converges. Otherwise it diverges. a Method 1. P-Test f(x) ≥ 0 and g(x) ≥ 0 and are both continuous from a