3.4 Class Notes - Prelim 2 results 99-95: A+ 94-82: A...

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Prelim 2 results 0 5 10 15 20 25 99-95: A+ 94-82: A 81-77: A- 76-72: B+ 70-59: B 58-54: B- 53-49: C+ 48-36: C 35-30: C- 28-10: D Integration by parts Product Rule ( )' ' ' ' ' Sometimes written as (, ' ) ' fg d x fg f fg f g f g fg f g dx f g dx uf gd dv g dx udv v vdu x u =+ == =− ∫∫ x f g dx fg f g dx () , ' '( ) 1 , (1 ) ) ) ( x xx x x x x x xe dx xe e dx x e xe e fx xgx e fx gx e xe = −= + = x f x f g fgd x 2 2 22 2 2( 2 ( 1 ) ) (2 2 ) ' ( 2 2 )(2 2 ) , ' '( ) 2 , ) ( x x x x x x xedx x x e xxe x fx x gx e ex = −+ = − + −+ = = = = = x 3 x 32 23 2 Guess ( ) . Derivative: (3 2 ) ( ) Want 3, 2 6, 6 Answer ( 3 6 6) Can find for 1,2,3,4,. .. x x x x px xa xb x c e x ax b e x ax bx c e ab a c b xxxe x e dx p ++ + + + = =− =− = IBP kills logarithms 2 2 11 1 2 2 () l n, ' '( ) ln 1 ln ( ) ln 1 ANS ln 1/ , ( ) x gx x xgx x dx x d x x x x d dx x x x x = + =
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Where is the other function? () l n, ' () 1 '( ln 1 ln ln 1 A )1 /, NS ) ln 1 ( xdx xx x d x xxx x d dx fx x gx xgx x x =− == =+ = = Definite integrals () 1 1 1 1 ' '( ) 1/ , 1 ln ln ANS=( ) (1ln1 0) ( 1 ) e e e e x d x xxx x ee = = −− −= = Problem 40. Rumen Fermentation 1 3 0 1 1 33 0 0 1 3 3 3 0 () (1 ) , ' () 3 '( ) 1 3( 1 ) (1 ) 1 ) 1 3 ,( ) t tt t t t t f g te ft et d t ete d t e gt = ⎛⎞ − + = −−+ ⎜⎟ ⎝⎠ = New ways for old problems 1/2 3/2 5/2 , () ( 1 ) 2 '( ) 1, ( ) ) 3 1 22 ) ) 2 ) ) 5 fx xgx x gx d x x d x x x + − ⋅ The old way (substitution)
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3.4 Class Notes - Prelim 2 results 99-95: A+ 94-82: A...

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