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Unformatted text preview: (8.8) Improper Integrals Infinite integrals. The definition of ∞ a f ( x ) dx is ∞ a f ( x ) dx = lim t →∞ t a f ( x ) dx. Put differently, suppose F ( x ) is an antiderivative of f ( x ), so f ( x ) dx = F ( x ) + C . So t a f ( x ) dx = F ( x ) t a = F ( t ) F ( a ). Then ∞ a f ( x ) dx = lim t →∞ ( F ( t ) F ( a ) ) . Example. ∞ e 5 x dx . Here, t e 5 x dx = 1 5 e 5 x t = 1 5 1 5 e 5 t . So ∞ e 5 x dx = lim t →∞ t e 5 x dx = lim t →∞ ( 1 5 1 5 e 5 t ) = 1 5 . (Note lim t →∞ e 5 t = lim t →∞ 1 e 5 t = 0; this is because 1 large = small.) Example. ∞ x 2 e x dx . Here, by two integrations by parts, x 2 e x dx = ( x 2 + 2 x + 2) e x + C. Therefore, ∞ x 2 e x dx = lim t →∞ t x 2 e x dx = lim t →∞ ( x 2 + 2 x + 2) e x t = lim t →∞ ( t 2 + 2 t + 2) e t + 2 . We must compute this limit. Now lim t →∞ t 2 e t = lim t →∞ t 2 e t = lim t →∞ 2 t e t = lim t →∞ 2 e t = 0 by two applications of L’Hospital’s rule. In the same way, we see ∞ x 2 e x dx = lim t →∞ ( t 2 + 2 t + 2) e t + 2 = 2 . Note, it may be useful to remember that if n is any positive integer, then lim t →∞ t n e t = 0. (This requires n applications of L’Hospital’s rule. This says that e t grows much faster for large t than does t n .) 1 Example....
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 Fall '08
 VALDIMARSSON
 Calculus, Derivative, Improper Integrals, Integrals, dx, Riemann integral

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