improp_ints

improp_ints - (8.8) Improper Integrals Infinite integrals....

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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 LHospitals 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 LHospitals rule. This says that e t grows much faster for large t than does t n .) 1 Example....
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improp_ints - (8.8) Improper Integrals Infinite integrals....

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