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Unformatted text preview: MAC 2312 Homework 2 Selected Solutions 4. Find a function f ( x ) satisfying f ( x ) = radicalbig (1 + x ) / ( x 1) for < x < 1. Solution: In this problem, I wanted to be sure that in addition to knowing standard integration techniques, you were paying attention to every step of your solution. In particular, you need to realize that since < x < 1, radicalbig ( x + 1) 2 =  x + 1  = ( x + 1) . (1) Now, let f ( x ) denote a function whose derivative is radicalbig (1 + x ) / ( x 1). To find f , simply compute integraltext radicalbig (1 + x ) / ( x 1) dx . To compute this integral, start by using both equation (1) and linearity of integration to split the integral into two integrals as follows f ( x ) = integraldisplay radicalbigg 1 + x x 1 dx = integraldisplay radicalbigg 1 + x x 1 1 + x 1 + x dx = integraldisplay radicalbigg (1 + x ) 2 x 2 1 dx = integraldisplay x + 1 x 2 1 dx = I 1 I 2 , (2) where I 1 = integraldisplay x x 2 1 dx and I 2 = integraldisplay 1 x 2 1 dx. To handle I 1 , use the change of variable u = x 2 1 du = 2 x dx then perform standard integrations as follows I 1 = 1 2 integraldisplay 1 u du = u + C = x 2 1 + C. To handle I 2 , use the trigonometric substitution x = sec , dx = sec tan d ; < 3 2 . (3) With this substitution, we have x 2 1 = sec 2 1 = tan 2 =  tan  = tan , (4) 1 where the final equality holds since tan 0 for < 3 2 . Using both (3) and (4), we get I 2 = integraldisplay sec tan tan d = ln  sec + tan  + C (5) = ln  x + x 2...
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 Spring '08
 Bonner
 Calculus, Addition

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