solutionQ10A - Solutions to Quiz 10A...

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Unformatted text preview: Solutions to Quiz 10A www.math.ufl.edu/˜harringt November 27, 2006 1. (2 pts.) Find f (x) by solving the intial value problem. f (x) = 1+ex +3x2 ; f (1) = e. Solution: f (x) = f (x)dx = 1 + ex + 3x2 dx = x + ex + x3 + c where c is a constant. Now, we have to find the value for c. So, f (1) = 1 + e1 + 13 + c = 2 + e + c. Recall: f (1) = e. Hence, 2+ e + c = e implies that c = −2. So f (x) = x + ex + x3 − 2. ln x dx. x 2. (2 pts.) Find the indefinite integral. 1 Solution: Here we have to use u subsitution. Let u = ln x, so du = x dx. So we have: ln x dx = x 1 ln(x) ∗ dx x u = du udu 12 u +C 2 1 (ln(x))2 + C = 2 = 1 Thus, ln x dx = 2 (ln(x))2 + C where C is a constant. x As a reminder, make sure you keep this fact straight: ln x2 = (ln x)2 . 3. (1 pt.) True or False. If f and g are integrable, then f (x)g (x)dx = f (x)dx g (x)dx. Solution: This is a false statement. Let f (x) = 1 and g (x) = 1. f (x)g (x)dx = f (x)dx g (x)dx = 1dx = x + c0 1dx 1dx = (x + c1 )(x + c2 ) = x2 + c1 x + c2 x + c1 c2 So, f (x)g (x)dx = f (x)dx g (x)dx when f (x) = 1 and g (x) = 1. Note: each time we integrate, we may obtain different constant terms. In other words, c0 , c1 and c2 may be a different numbers. 1 ...
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