Homework 2 Solutions

Homework 2 Solutions - MAC 2312 Homework 2 Selected...

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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 θ, dx = sec θ tan θ dθ ; π ≤ θ < 3 π 2 . (3) With this substitution, we have √ x 2 − 1 = √ sec 2 θ − 1 = √ tan 2 θ = | tan θ | = tan θ, (4) 1 where the final equality holds since tan θ ≥ 0 for π ≤ θ < 3 π 2 . Using both (3) and (4), we get I 2 = integraldisplay sec θ tan θ tan θ dθ = ln | sec θ + tan θ | + C (5) = ln | x + √ x 2 −...
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Homework 2 Solutions - MAC 2312 Homework 2 Selected...

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