Calculus Cheat Sheet.pdf

# Improper integral an improper integral is an integral

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Improper Integral An improper integral is an integral with one or more infinite limits and/or discontinuous integrands. Integral is called convergent if the limit exists and has a finite value and divergent if the limit doesn’t exist or has infinite value. This is typically a Calc II topic. Infinite Limit 1. ( ) ( ) lim t a a t f x dx f x dx →∞ = 2. ( ) ( ) lim b b t t f x dx f x dx →−∞ = 3. ( ) ( ) ( ) c c f x dx f x dx f x dx = + provided BOTH integrals are convergent. Discontinuous Integrand 1. Discont. at a : ( ) ( ) lim b b a t t a f x dx f x dx + = 2. Discont. at b : ( ) ( ) lim b t a a t b f x dx f x dx = 3. Discontinuity at a c b < < : ( ) ( ) ( ) b c b a a c f x dx f x dx f x dx = + provided both are convergent. Comparison Test for Improper Integrals : If ( ) ( ) 0 f x g x on [ ) , a then, 1. If ( ) a f x dx conv. then ( ) a g x dx conv. 2. If ( ) a g x dx divg. then ( ) a f x dx divg. Useful fact : If 0 a > then 1 a p x dx converges if 1 p > and diverges for 1 p . Approximating Definite Integrals For given integral ( ) b a f x dx and a n (must be even for Simpson’s Rule) define b a n x = and divide [ ] , a b into n subintervals [ ] 0 1 , x x , [ ] 1 2 , x x , … , [ ] 1 , n n x x with 0 x a = and n x b = then, Midpoint Rule : ( ) ( ) ( ) ( ) * * * 1 2 b n a f x dx x f x f x f x ≈ ∆ + + + L , * i x is midpoint [ ] 1 , i i x x Trapezoid Rule : ( ) ( ) ( ) ( ) ( ) ( ) 0 1 2 1 2 2 2 2 b n n a x f x dx f x f x f x f x f x + + + + + + L Simpson’s Rule : ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 2 2 1 4 2 2 4 3 b n n n a x f x dx f x f x f x f x f x f x + + + + + + L
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