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AP Calculus Integration U-Substitution & Properties Laura Habberfield Per 4 When you take derivatives, you frequently have to use the chain rule to differentiate. The integration equivalent of the chain rule is called u-substitution. x( x 2 1)5 dx ( x 2 1) Set up a u = _______ Find du 2x dx = _______. 2 x dx Solve for du = _______ You need to manufacture your du in the original expression. So you will have to multiply by _____ on the inside and thus 2 1 multiply by ______ on the outside. 2 Now change everything to u. u 5du 1 u6 Now integrate in terms of u. 6 Finally, change back to the variable x and add C. 1 ( x 2 1) 6 C 6 http://www.mastermathmentor.com/mmm/calc/default.aspx?page=ABManual Integral Properties a a f ( x ) dx If we start at a and end at a, there is no area. b a f ( x) dx f ( x) dx From a to b gives an area. From b to a gives the negative of this area. a b b c c The area from a to b plus the area a f ( x) dx f ( x) dx f ( x) dx b a from b to c = the area from a to c when f(x) is continuous on the integral [a,c] http://www.mastermathmentor.com/mmm/calc/default.aspx?page=ABManual Integral Properties (continued) # # If the integral is symmetric in the y-axis, its limits 2 f ( x) dx f ( x) dx can be from 0 to a # and multiplied by 2, or from 0 # the negative number to the positive number. f ( x) dx 0 If the integral is symmetric in the origin, it will always = 0. f ( x) dx 2 f ( x) dx 2 The integral of two functions or lines is equal to the integral of each piece alone. Using integrals numerically is generally done with Riemann Sums or the Trapezoid Rule, which are both covered under another review section. YOUR TURN!!! Try these multiple choice questions on what you just reviewed! Answers follow. 1. If , and , find the value of A. 3 B. -3 C. 2 D. -2 E. Can’t be determined http://webs.bcp.org/sites/jmolina/summer/Indefinite%20a nd%20definite%20integrals,%20and%20u-substitution.htm 2. Find the value of the integral A. B. C. D. E. e2cos x sin x dx 2 3. 0 A. e2 e B. 1 C. e2 D. 0 E. Does not exist Calculus AB/BC http://webs.bcp.org/sites/jmolina/summer/Indefinite%20a nd%20definite%20integrals,%20and%20u-substitution.htm Maxine Lifshitz 4. Let f(x) be an odd function and g(x) be even. A. II Which of the following statements are true? B. III 2 I. 2 f ( x) dx 0 C. I and II II. D. I and III 2 2 g ( x) dx 0 2 III. 2 f ( x) g ( x) dx 0 E. I, II, and III 5. Find x 2 x 3 1 dx 5 Answers ( x 3 1) 6 1. A C A. 18 2. C ( x 3 1) 6 B. C 3. A 6 C. ( x 3 1) 6 4. D C 2 5. A D. 6( x 3 1) 6 C E. 2 x( x 3 1)5 C Calculus AB/BC Maxine Lifshitz 1. Traffic flow is determined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular interesection is modeled by the function F defined by: t F (t ) 82 4 sin for 0 t 30 2 Set up an integral, but do not solve, to determine how many cars pass through the intersection over the 30-minute period. 30 t 0 82 4 sin 2 http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html?calcab 2. At an intersection in Thomasville, Oregon, cars turn left at the rate t L(t ) 60 t sin 2 3 cars per hour over the time interval 0<t<18 hours. The graph of y=L(t) is shown below. To the nearest whole number, find the total number of cars turning left at the intersection over the time interval 0<t<18 hours. Calculator available. 18 t 60 t sin dt 1658 cars 2 0 3 http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html?calcab