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Unformatted text preview: Fall 2011 MA 16200 Study Guide - Exam # 2 (1) Integration via Partial Fractions : Use for (proper) rational functions R ( x ) Q ( x ) ; If degree R ( x ) degree Q ( x ), i.e. rational function is improper, then do long division before using partial fractions. (2) Integration via Clever Substitutions/Using Integral Tables : Use a substitution to transform integral into a form to be able to use another integral techniques ( Substitution Method, Integration by Parts, Trig Integrals, Trig Subsitution Method , etc) or use inte- gral tables. (3) Approximating definite integrals integraldisplay b a f ( x ) dx . Let x = b a n , x k = a + k x and x k = 1 2 ( x k 1 + x k ) (Note that x = a and x n = b ) (a) Midpoint Rule : integraldisplay b a f ( x ) dx M n = ( x ) [ f ( x 1 ) + f ( x 2 ) + + f ( x n )] (b) Trapezoidal Rule : integraldisplay b a f ( x ) dx T n = parenleftbigg x 2 parenrightbigg [ f ( x ) + 2 f ( x 2 ) + 2 f ( x 3 ) + + 2 f ( x n 1 ) + f ( x n )] (c) Simpsons Rule : Only works for n even ....
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