Calculus_Cheat_Sheet_Integrals

Calculus_Cheat_Sheet_Integrals - Calculus Cheat Sheet...

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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Integrals Definitions Definite Integral: Suppose () fx is continuous on [ ] , ab . Divide [ ] , into n subintervals of width x Δ and choose * i x from each interval. Then () () * 1 lim i b a n i f x dx f x x →∞ = . Anti-Derivative : An anti-derivative of ( ) is a function, ( ) Fx , such that () ( ) Fx fx = . Indefinite Integral : f x dx F x c =+ where ( ) is an anti-derivative of ( ) . Fundamental Theorem of Calculus Part I : If is continuous on [ ] , then x a gx ftd t = is also continuous on [ ] , and x a d t fx dx == . Part II : is continuous on [ ] , , ( ) is an anti-derivative of ( i.e. fxd x = ) then () () () b a x Fb Fa =− . Variants of Part I : ux a d f t dt u x f u x dx =⎡ ⎣⎦ b vx d f t dt v x f v x dx [] d t uxf vxf dx ′′ Properties () () () () f x g x dx f x dx g x dx ±= ± ∫∫ bb b aa a f x g x dx f x dx g x dx ± 0 a a x = ba x x ( ) ( ) cf x dx c f x dx = , c is a constant cf x dx c f x dx = , c is a constant f x dx f t dt = x x If fx gx on axb ≤≤ then f x dx g x dx If 0 on then 0 b a x If mfx M on then ( ) b a mb a f xd x M b a −≤ Common Integrals kdx kx c 1 1 1 ,1 nn n xd x x cn + + 1 1 ln x x d x x c + 11 ln a ax b dx ax b c + + ln ln udu u u u c + uu du c ee cos sin udu sin cos + 2 sec tan sec tan sec d u u c csc cot csc u udu u c + 2 csc cot + tan ln sec u c sec ln sec tan u u c + 1 1 1 22 tan u au du c + 1 1 sin u a du c
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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Standard Integration Techniques Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class. u Substitution : The substitution ( ) ug x = will convert () () () ( ) bg b ag a fgx gxd x fud u = ∫∫ using du g x dx = . For indefinite integrals drop the limits of integration. Ex. 23 2 1 5c o s xx d x 32 2 1 3 3 u x du x dx x dx du =⇒ = = 33 11 1 : : 22 8 xu =⇒== ( ) 28 5 3 8 55 1 5 cos cos sin sin 8 sin 1 x x dx u dx u = == Integration by Parts : udv uv vdu =− and bb b a aa . Choose u and dv from integral and compute du by differentiating u and compute v using vd v = . Ex. x xd x e u x dv du dx v −− = = ee x x x d x x c + + e Ex. 5 3 ln xdx 1 ln x u x dv dx du dx v x = =⇒= = 5 5 3 3 ln ln ln 5 l n5 3 l n3 2 x x dx x x x = =−− Products and (some) Quotients of Trig Functions For sin cos nm d x we have the following : 1. n odd. Strip 1 sine out and convert rest to cosines using sin 1 cos , then use the substitution cos ux = .
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Calculus_Cheat_Sheet_Integrals - Calculus Cheat Sheet...

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