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 ux v x x du dx dv dx x == ( ) 22 2 11 1 ln ln | 2 ln ln | 2 ln | 2 ee 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 = ) 2. Evaluate: 2 2 1 32 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 1 1 1 2 2 3 214 4 21 41 dx dx dx dx dx xx x = = −−+ −− + + + . Let 12 s i n x θ −= . Then 2cos dx d θθ = . 1 1 sin 2 44 s i n 41 dx x dd C C x ⎛⎞ = + = + ⎜⎟ ⎝⎠ () () 1 1 1 1 26 1 sin | sin sin 0 2 x π =−=
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3. Evaluate: 2 tan sec x xdx Let tan ux = . Then 2 sec du x dx = () 3 3 2 2 2 22 33 tan sec tan x udu u C x C == + = + ∫∫ 4. Evaluate: dx x x 1 tan 2 1 2 tan 2 1 1 x v du dx dv x dx x + 2 2 11 1 1 1 1 1 tan tan ( ) tan ( ) 1 tan ( ) tan 1 1 xx x x x xd x x d x x d x x x x C −− ⎛⎞ =− + ⎜⎟ ++ ⎝⎠
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Test 1 Fall 2006 _corrected_ _solutions_ - Math 1B First...

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