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Chapter2UsefulResults

Chapter2UsefulResults - f ne Z ∞ −∞ f x dx = Z a...

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First Fundamental Theorem of Calculus (FTCI) If f is continuous on [ a, b ] then the function g de fi ned by g ( x ) = Z x a f ( t ) dt, a x b is continuous on [ a, b ] and di ff erentiable on ( a, b ) and g 0 ( x ) = f ( x ) . FTCI and the Chain Rule Suppose we want the derivative with respect to x of G ( x ) where G ( x ) = Z h ( x ) a f ( t ) dt, a x b and h ( x ) is a di ff erentiable function on [ a, b ] . If we de fi ne g ( u ) = Z u a f ( t ) dt then G ( x ) = g ( h ( x )) . Then by the Chain Rule G 0 ( x ) = g 0 ( h ( x )) · h 0 ( x ) = f ( h ( x )) · h 0 ( x ) a < x < b. Exercise : Find G 0 ( x ) if G ( x ) = Z h 2 ( x ) h 1 ( x ) f ( t ) dt, a x b Hint: G ( x ) = Z c h 1 ( x ) f ( t ) dt + Z h 2 ( x ) c f ( t ) dt , a < c < b 1
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Improper Integrals ( a ) If R b a f ( x ) dx exists for every number b a then Z a f ( x ) dx = lim b →∞ Z b a f ( x ) dx provided this limit exists. If the limit exists we say the improper integral con- verges otherwise we say the improper integral diverges. ( b ) If R b a f ( x ) dx exists for every number a b then Z b −∞ f ( x ) dx = lim a →−∞ Z b a f ( x ) dx provided this limit exists. ( c ) If both R a f ( x ) dx and R a −∞ f ( x ) dx
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Unformatted text preview: f ne Z ∞ −∞ f ( x ) dx = Z a −∞ f ( x ) dx + Z ∞ a f ( x ) dx where a is any real number. Comparison Test for Improper Integrals Suppose that f and g are continuous functions with f ( x ) ≥ g ( x ) ≥ for x ≥ a . ( a ) If Z ∞ a f ( x ) dx is convergent then Z ∞ a g ( x ) dx is convergent. ( b ) If Z ∞ a g ( x ) dx is divergent then Z ∞ a f ( x ) dx is divergent . Useful Result for Using Comparison Test Z ∞ 1 1 x p dx converges if and only if p > 1 . Useful Inequalities: 1 1 + y p ≤ 1 y p , y ≥ 1 1 1 + y p ≥ 1 y p + y p = 1 2 y p , y ≥ 1 2...
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