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ImproperIntegrals

ImproperIntegrals - Improper Integrals Definitions I 1 t f...

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Improper Integrals

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Definitions I
R c where exist dx x f and dx x f tegrals in improper the of both provided dx x f dx x f dx x f diverge to said is tegral in the Otherwise mits li the of value the to converge to said is tegral in the case which in exist mits li these ovided dx x f dx x f dx x f dx x f c c c c b t b t t a a t + = = = + - + - + - - -∞ + ∞ + ∞ , ) ( ) ( ) ( ) ( ) ( . 2 . . ; Pr ) ( lim ) ( . 2 ) ( lim ) ( . 1

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Examples I
Example (1) 2 2 ) (arctan lim ] 0 [arctan lim ] 0 arctan [arctan lim ( lim 1 1 lim 1 1 ) arctan 0 0 2 0 2 π π = = = - = - = = + = + + ∞ + ∞ + ∞ + ∞ + ∞ + ∞ t t t dx x dx x t t t t t t t x

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Example (2) 2 ) 2 ( 0 ) (arctan lim ] arctan 0 [ lim ] arctan 0 [arctan lim ( lim 1 1 lim 1 1 ) arctan 0 0 2 0 2 π π = - - = - = - = - = = + = + -∞ -∞ -∞ -∞ - -∞ t t t dx x dx x t t t t t t t x
Example (3) π π π = + = + + + = + - + + ∞ - 2 2 1 1 1 1 1 1 0 0 2 2 2 dx x dx x dx x

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Example (4) diverges I Thus Why t t t x dx x dx I t t t t t t t x ?) ( ln lim ] 0 [ln lim ] 1 ln [ln lim ( lim lim ) ln 1 1 1 + ∞ = = - = - = = = = + ∞ + ∞ + ∞ + ∞ + ∞ + ∞
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