105-Mid1Sols - Math 105 Section 208 209 Midterm 1 Solutions...

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M ath 105, Section 208 & 209 M idterm 1 S olutions 1. (12 points) Compute the following integral: Z dx ( x - 1)( x + 2) 2. (13 points) Compute the following integral: Z 4 1 t ln t dt 3. (10 points) Find the net area and the total area between the graph of the function f ( x ) = 3 cos x and the x -axis on the interval [ π 2 , 2 π ]. 4. (3 × 5 points) Give short answers to the following questions: (a) Using Riemann sums, identify the following limit as a definite integral but do not evaluate the integral: lim n →∞ n X k =1 2 n cos 1 + 2 k n - 1 (b) Find the following derivative in terms of f : d dx Z arctan( x 2 ) 0 f ( t ) dt (c) Find the trapezoidal rule approximation of Z 2 0 ( x 2 - 1 ) dx with n = 2. 1
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1. Compute Z dx ( x - 1)( x + 2) . The best thing to do here is substitute u = x, du = 1 2 x dx dx = 2 xdu = 2 udu. This gives Z dx ( x - 1)( x + 2) = 2 Z u ( u - 1)( u + 2) du. Using partial fractions we get u ( u - 1)( u + 2) = 1 3 1 u - 1 + 2 3 1 u + 2 , hence 2 Z u ( u - 1)( u + 2) du = 2 3 Z 1 u - 1 du + 4 3 Z 1 u + 2 du = 2 3 ln | u - 1 | + 4 3 ln | u + 2 | + C = 2 3 ln | x - 1 | + 4 3 ln | x + 2 | + C. A lot of students tried to use partial fractions without doing the substitution first. This is not wrong, and does lead to the answer, but it’s harder than the solution above. Be aware that partial fractions is only recommended for fractions of polynomials , so not when there are x ’s.
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