Cheatsheet for Calculus Derivatives to Integration, Cheat Sheet of Mathematics

Download Cheatsheet for Calculus Derivatives to Integration and more Cheat Sheet Mathematics in PDF only on Docsity! Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Integrals Definitions Definite Integral: Suppose ( )f x is continuous on [ ],a b . Divide [ ],a b into n subintervals of width x∆ and choose * ix from each interval. Then ( ) ( )* 1 lim i nb a n i f x dx f x x →∞ = = ∆∑∫ . Anti-Derivative : An anti-derivative of ( )f x is a function, ( )F x , such that ( ) ( )F x f x′ = . Indefinite Integral : ( ) ( )f x dx F x c= +∫ where ( )F x is an anti-derivative of ( )f x . Fundamental Theorem of Calculus Part I : If ( )f x is continuous on [ ],a b then ( ) ( ) x a g x f t dt= ∫ is also continuous on [ ],a b and ( ) ( ) ( ) x a dg x f t dt f x dx ′ = =∫ . Part II : ( )f x is continuous on[ ],a b , ( )F x is an anti-derivative of ( )f x (i.e. ( ) ( )F x f x dx= ∫ ) then ( ) ( ) ( ) b a f x dx F b F a= −∫ . Variants of Part I : ( )( ) ( ) ( ) u x a d f t dt u x f u x dx ′=   ∫ ( ) ( ) ( ) ( ) b v x d f t dt v x f v x dx ′= −   ∫ ( ) ( ) ( ) ( ) [ ] ( ) [ ]( ) ( ) u x v x u x v x d f t dt u x f v x f dx ′ ′= −∫ Properties ( ) ( ) ( ) ( )f x g x dx f x dx g x dx± = ±∫ ∫ ∫ ( ) ( ) ( ) ( ) b b b a a a f x g x dx f x dx g x dx± = ±∫ ∫ ∫ ( ) 0 a a f x dx =∫ ( ) ( ) b a a b f x dx f x dx= −∫ ∫ ( ) ( )cf x dx c f x dx=∫ ∫ , c is a constant ( ) ( ) b b a a cf x dx c f x dx=∫ ∫ , c is a constant ( ) b a c dx c b a= −∫ ( ) ( ) b b a a f x dx f x dx≤∫ ∫ ( ) ( ) ( ) b c b a a c f x dx f x dx f x dx= +∫ ∫ ∫ for any value of c. If ( ) ( )f x g x≥ on a x b≤ ≤ then ( ) ( ) b b a a f x dx g x dx≥∫ ∫ If ( ) 0f x ≥ on a x b≤ ≤ then ( ) 0 b a f x dx ≥∫ If ( )m f x M≤ ≤ on a x b≤ ≤ then ( ) ( ) ( ) b a m b a f x dx M b a− ≤ ≤ −∫ Common Integrals k dx k x c= +∫ 11 1 , 1n n nx dx x c n+ += + ≠ −∫ 1 1 lnxx dx dx x c− = = +∫ ∫ 1 1 lnaa x b dx ax b c + = + +∫ ( )ln lnu du u u u c= − +∫ u udu c= +∫e e cos sinu du u c= +∫ sin cosu du u c= − +∫ 2sec tanu du u c= +∫ sec tan secu u du u c= +∫ csc cot cscu udu u c= − +∫ 2csc cotu du u c= − +∫ tan ln secu du u c= +∫ sec ln sec tanu du u u c= + +∫ ( )11 1 2 2 tan u a aa u du c− + = +∫ ( )1 2 2 1 sin u aa u du c− − = +∫ 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 ( )u g x= will convert ( )( ) ( ) ( ) ( ) ( )b g b a g a f g x g x dx f u du′ =∫ ∫ using ( )du g x dx′= . For indefinite integrals drop the limits of integration. Ex. ( )2 32 1 5 cosx x dx∫ 3 2 2 1 33u x du x dx x dx du= ⇒ = ⇒ = 3 31 1 1 :: 2 2 8x u x u= ⇒ = = = ⇒ = = ( ) ( ) ( ) ( ) ( )( ) 2 32 8 5 31 1 85 5 3 31 5 cos cos sin sin 8 sin 1 x x dx u du u = = = − ∫ ∫ Integration by Parts : u dv uv v du= −∫ ∫ and b bb aa a u dv uv v du= −∫ ∫ . Choose u and dv from integral and compute du by differentiating u and compute v using v dv= ∫ . Ex. xx dx−∫ e x xu x dv du dx v− −= = ⇒ = = −e e x x x x xx dx x dx x c− − − − −= − + = − − +∫ ∫e e e e e Ex. 5 3 ln x dx∫ 1ln xu x dv dx du dx v x= = ⇒ = = ( )( ) ( ) ( ) 5 5 55 3 33 3 ln ln ln 5ln 5 3ln 3 2 x dx x x dx x x x= − = − = − − ∫ ∫ Products and (some) Quotients of Trig Functions For sin cosn mx x dx∫ we have the following : 1. n odd. Strip 1 sine out and convert rest to cosines using 2 2sin 1 cosx x= − , then use the substitution cosu x= . 2. m odd. Strip 1 cosine out and convert rest to sines using 2 2cos 1 sinx x= − , then use the substitution sinu x= . 3. n and m both odd. Use either 1. or 2. 4. n and m both even. Use double angle and/or half angle formulas to reduce the integral into a form that can be integrated. For tan secn mx x dx∫ we have the following : 1. n odd. Strip 1 tangent and 1 secant out and convert the rest to secants using 2 2tan sec 1x x= − , then use the substitution secu x= . 2. m even. Strip 2 secants out and convert rest to tangents using 2 2sec 1 tanx x= + , then use the substitution tanu x= . 3. n odd and m even. Use either 1. or 2. 4. n even and m odd. Each integral will be dealt with differently. Trig Formulas : ( ) ( ) ( )sin 2 2sin cosx x x= , ( ) ( )( )2 1 2cos 1 cos 2x x= + , ( ) ( )( )2 1 2sin 1 cos 2x x= − Ex. 3 5tan secx x dx∫ ( ) ( ) ( ) 3 5 2 4 2 4 2 4 7 51 1 7 5 tan sec tan sec tan sec sec 1 sec tan sec 1 sec sec sec x xdx x x x xdx x x x xdx u u du u x x x c = = − = − = = − + ∫ ∫ ∫ ∫ Ex. 5 3 sin cos x x dx∫ ( ) 2 21 1 2 2 2 25 4 3 3 3 2 2 3 2 2 2 4 3 3 sin(sin )sin sin sin cos cos cos sin(1 cos ) cos (1 ) 1 2 cos sec 2ln cos cos xxx x x x x x xx x u u u u u dx dx dx dx u x du du x x x c − − − + = = = = = − = − = + − + ∫ ∫ ∫ ∫ ∫ ∫