ch8sc7notes - Math 104 Rimmer 8.7 Improper Integrals Infinite Upper Limit b f x dx = lim f x dx b a 1 b b e a 2 x dx = lim e b 2 x 1 b 1 2 x 1 lim lim

ch8sc7notes - Math 104 Rimmer 8.7 Improper Integrals...

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1 Math 104 – Rimmer 8.7 Improper Integrals ( ) ( ) lim b b a a f x dx f x dx →∞ = 2 1 x e dx - 2 1 lim b x b e dx - →∞ = 2 1 1 lim 2 b x b e - →∞ - = 2 1 2 e = 2 2 1 1 lim 2 2 b b e e →∞ - = + 2 1 1 lim 2 b x b e →∞ - = 2 1 since lim 0 2 b b e →∞ - = Infinite Upper Limit Math 104 – Rimmer 8.7 Improper Integrals 1 1 x x e dx e + 1 lim 1 b x x b e dx e →∞ = + ( ) 1 lim ln 1 b x b e →∞ = + = DIVERGENT Infinite Upper Limit 1 x x u e du e dx = + = 1 ln u du u C = + ( ) ( ) limln 1 ln 1 b b e e →∞ = + - +
2 Math 104 – Rimmer 8.7 Improper Integrals ( ) ( ) lim b b a a f x dx f x dx →-∞ -∞ = 1 x xe dx -∞ 1 lim x a a xe dx →-∞ = 1 lim x x a a xe e →-∞ = - ( ) ( ) lim a a a e e ae e →-∞ = - - - Infinite Lower Limit x D I x e 1 x e 0 x e + - ( ) lim 1 a a e a →-∞ = - 0 (indeterminate) = ⋅∞ 1 lim a a a e - →-∞ - = ( ) L'Hospital = ' 1 lim L H a a e - →-∞ - = - 0 = Math 104 – Rimmer 8.7 Improper Integrals ( ) ( ) ( ) ; any real number c c f x dx f x dx f x dx c -∞ -∞ = + 3 π = Infinite Upper and Lower Limit ( ) ( ) ( ) lim lim c b a b a c f x dx f x dx f x dx →-∞ →∞ -∞ = + 2 6 1 x dx x -∞ + 0 2 2 6 6 0 lim lim 1 1 b a b a x x dx dx x x →-∞ →∞ = + + + 3 2 2 1 3 3 u x du x dx du x dx = = = 2 1 1 1 3 3 1 arctan u du u C + = + ( ) ( ) 0 3 3 1 1 3 3 0 lim arctan lim arctan b a b a x x →-∞ →∞ = + ( ) ( ) 3 3 1 1 3 3 lim arctan lim arctan a b a b →-∞ →∞ = - + ( ) ( ) 1 1 3 2 3 2 π π = - - +
3 Math 104 – Rimmer 8.7 Improper Integrals Infinite Discontinuity at Lower Limit ( ) ( ) ( ) infinite discontinuity lim b b t a a t f a f x dx f x dx + = ( ) 9 0 0 infinite discontinuity f dx x 9 0 lim t t dx x + = 9 1/2 0 lim t t x dx + - = 9 1/ 2 0 lim 2 t t x + = 9 0 lim 2 t t x + = 0 lim 6 2 t t + = - 6 = Math 104 – Rimmer 8.7 Improper Integrals Infinite Discontinuity at Upper Limit ( ) ( ) ( ) infinite discontinuity lim b t t b a a f b f x dx f x dx - = ( ) 8 3 0 8

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