Midterm2Dsol - Math 125D Midterm II Solutions 1 9 points...

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Unformatted text preview: Math 125D Midterm II Solutions November 15, 2007 1. [ 9 points total ] Evaluate the following integral: Z 2 e x e 2 x- 3 e x + 2 dx Solution. Take u = e x , du = e x dx . Z 2 e x e 2 x- 3 e x + 2 dx = Z 2 u 2- 3 u + 2 du . We can factor the denominator as u 2- 3 u + 2 = ( u- 2)( u- 1) . Using partial fractions, we compute Z 2 u 2- 3 u + 2 du = 2 Z 1 u- 2- 1 u- 1 du = 2ln fl fl fl fl e x- 2 e x- 1 fl fl fl fl + C 2. [ 9 points total ] Evaluate the following definite integral: π/ 6 Z cos 2 (3 x )sin 3 (3 x ) dx Solution. Write π/ 6 Z cos 2 (3 x )sin 3 (3 x ) dx = π/ 6 Z cos 2 (3 x )sin 2 (3 x )sin(3 x ) dx and take u = cos(3 x ). Then du =- 1 3 sin(3 x ) dx . The integral becomes- 1 3 Z 1 u 2 (1- u 2 ) du = 1 3 Z 1 u 2- u 4 du = 2 45 3. [ 9 points total ] Evaluate the following integral: Z x 3 √ x 2- 1 dx Solution. Make a u-substitution u = x 2- 1. Then du = 2 xdx . Z x 3 √ x 2- 1 dx = 1 2 Z u + 1 √ u du = 1 2 Z u 1 / 2 + u- 1 / 2 du = 1 3 u 3 / 2 + u 1 / 2 = ( x 2- 1) 3 / 2 3 +...
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Midterm2Dsol - Math 125D Midterm II Solutions 1 9 points...

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