mt1-ans - Math 1B. Sample Answers to First Midterm 1. (10...

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Unformatted text preview: Math 1B. Sample Answers to First Midterm 1. (10 points) (a). Find Z x tan- 1 xdx . Use integration by parts with u = tan- 1 x dv = xdx du = dx x 2 +1 v = x 2 2 : Z x tan- 1 xdx = x 2 2 tan- 1 x- 1 2 Z x 2 x 2 + 1 dx = x 2 2 tan- 1 x- 1 2 Z 1- 1 x 2 + 1 dx = x 2 2 tan- 1 x- x 2 + 1 2 tan- 1 x + C . (b). Find Z xdx x 2- 3 x 2 . Make the rationalizing substitution u = 3 x , viewed as the inverse substitution x = u 3 , dx = 3 u 2 du , and then do the substitution v = u 4- 1, dv = 4 u 3 du : Z xdx x 2- 3 x 2 = Z u 3 3 u 2 du u 6- u 2 = Z 3 u 3 du u 4- 1 = 3 4 Z dv v = 3 4 ln | v | + C = 3 4 ln | u 4- 1 | + C = 3 4 ln 3 x 4- 1 + C . 2. (10 points) (a). Find Z x 2- 1 x dx . Do the trigonometric substitution x = sec , dx = sec tan d . Here p x 2- 1 = p sec 2 - 1 = p tan 2 = tan 1 2 (or draw a diagram). We have Z x 2- 1 x dx = Z tan sec tan d sec = Z tan 2 d = Z (sec 2 - 1) d = tan - + C = p x 2- 1- sec- 1 x + C ....
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mt1-ans - Math 1B. Sample Answers to First Midterm 1. (10...

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