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Test 1 Fall 2006 _corrected_ _solutions_

Test 1 Fall 2006 _corrected_ _solutions_ - Math 1B First...

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Math 1B First Exam Name:_________________________ SID:_____________________ (last, first) (Each problem worth 10 points) 1. Evaluate: ( ) e dx x 1 2 ln ( ) 2 ln 2ln u x v x x du dx dv dx x = = = = ( ) ( ) ( ) ( ) 2 2 2 1 1 1 1 1 ln ln | 2 ln ln | 2 ln | 2 e e e e e x dx x x x dx x x x x x e = = = (We got ln ln x dx x x x C = + by using integration by parts with ln u x = ) 2. Evaluate: 2 2 1 3 2 dx x x + Note: Originally, the limits were from 1 to 3. This makes the integral improper and I only wanted to test you on trigonometric substitution here). ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 1 1 1 1 1 2 2 2 3 2 2 3 2 1 4 4 2 1 4 1 dx dx dx dx dx x x x x x x x x x = = = = + + + + + . Let ( ) 1 2sin x θ = . Then 2cos dx d θ θ = . ( ) 1 2 2 2cos 1 sin 2 4 4sin 4 1 dx x d d C C x θ θ θ θ θ = = = + = + ( ) ( ) 1 2 1 1 1 1 2 6 1 sin | sin sin 0 2 x π = =
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