07 - FTC - a b ! 2. c z dz a b ! 3. z c dz a b ! 4. 1 4...

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Fundamental Theorem of Calculus Let g ( x ) = t dt 2 x ! Fundamental Theorem of Calculus (part 1): If f is continuous on [a, b], then the function defined by g ( x ) = f ( t ) dt a x ! , a " x " b is continuous on [a, b] and differentiable on (a, b) and ! g ( x ) = f ( x ).
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Ex. g ( x ) = cos t dt ! " x # Ex. g ( x ) = cos t dt x ! # Ex. g ( x ) = cos t dt ! x 2 #
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Ex h ( x ) = ln t dt 1 2 cos x ! Do. Find 1. d da cos u 2 v ( ) 3 a v ! du 2. d dc cos uv 2 ( ) " 2 c u ! dv 3. d dz cos uv 2 ( ) " 2 tan z ! dv
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Fundamental Theorem of Calculus (part 2) If f is continuous on [a, b], then f ( x ) dx a b ! = F ( b ) " F ( a ) where F is any antiderivative of f. ( ! F = f ) Ex. t t + t 2 3 dt 1 64 ! Ex. 3 x dx e e 3 !
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Integrate: 1. cz dz
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Unformatted text preview: a b ! 2. c z dz a b ! 3. z c dz a b ! 4. 1 4 " z 2 dz a b ! 5. e cz dz a b ! 6. cz e dz a b ! 7. 1 cz dz a b ! Ex. 2 x 6 dx ! 4 2 " Ex. 1 4 ! y dy 3 4 " Ex. f ( x ) dx ! " # , f ( x ) = x !" $ x $ sin x $ x $ " % & Do: 1. Which of the following integrals are improper? A . tan t + 1 ( ) dt ! 1 1 " B . 1 r 2 + 6 r + 8 dr ! 2 " C . ln 2 s ! 6 ( ) ds 4 9 " 2. Evaluate sec 2 z + b z + c 1 ! z 2 dz 1 2 "...
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07 - FTC - a b ! 2. c z dz a b ! 3. z c dz a b ! 4. 1 4...

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