Spring 2004 - Nordgren's Class - Exam 2

# Spring 2004 - Nordgren's Class - Exam 2 - Solutions to...

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Solutions to midterm 2 By H˚ akan Nordgren Question 1: Evaluate the integral R xe - 2 x dx . Answer: When the integrand consists of an x or x 2 (or x 3 , etc) multiplying a function like sin x , cos x or e x , then more likely than not, the best way to do the integral is to use integration by parts. So let’s try that. Let u = x and dv = e - 2 x dx . Then du = dx and v = - 1 2 e - 2 x . Thus we have Z xe - 2 x dx = x - 1 2 e - 2 x - Z - 1 2 e - 2 x dx = x - 1 2 e - 2 x - 1 4 e - 2 x dx + constant . Question 2: Use partial fractions to ﬁnd R 3 2 dx x 2 - 1 . Answer: x 2 - 1 = ( x - 1)( x + 1), so we can use partial fractions to express dx x 2 - 1 in the form 1 x 2 - 1 = A ( x - 1) + B ( x + 1) . Multiplying both sides of the above equation by x 2 - 1 = ( x - 1)( x + 1) we obtain 1 = A ( x - 1)( x + 1) ( x - 1) + B ( x - 1)( x + 1) ( x + 1) = A ( x + 1) + B ( x - 1) . We now want to ﬁnd A and B . The equation holds for every x , so we can pick x which make calculating A and B easy. First, let us consider the equation when

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Spring 2004 - Nordgren's Class - Exam 2 - Solutions to...

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