Transcription of Calculus Cheat Sheet Integrals - Lamar University
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Calculus Cheat Sheet Integrals - Lamar University
Calculus Cheat Sheet Visit for a complete set of Calculus notes. 2005 Paul Dawkins Integrals Definitions Definite Integral: Suppose ()fx is continuous on [ ],ab. Divide [ ],ab into n subintervals of width x a nd choose *ix from each interval. Then ()()*1liminbanif x dxf xx == . Anti-Derivative : A n anti-derivative of ()fx is a function, ( )Fx, such that ( ) ( )F x fx =. Indefinite Integral :( )( )f x dxF xc=+ where ( )Fx is an anti-derivative of ( )fx. Fundamental Theorem of Calculus Part I : If ( )fx is continuous on [],ab then ( )( )xag xf t dt= is also continuous on [ ],ab and ( )( )()xadg xf t dtf xdx == . Part II : ( )fxis continuous on[ ],ab, ( )Fx is an anti-derivative of( )fx( ( )( )F xf x dx= ) then( )( )()baf x dxF bF a= .Variants of Part I : ()( )()()uxadf t dtu x f u xdx = ( )( )( )( )bvxdf t dtv x f v xdx = ( )( )( )( )[ ]( )[ ]()()uxvxuxvxdftdt u xfv xfdx = Properties ( ) ( )( )( )f xg x dxf x dxg x dx = ( ) ( )( )()bbbaaaf xg x dxf x dxg x dx = ()0aaf x dx= ( )( )baabf x dxf x dx= ( )( )cf x dxc f x dx= , c is a constant ()()bbaacf x dxcf x dx= , c is a constant ()bac dxc b a= ( )( )bbaaf x dxf x dx ( )( )( )b cba acf x dxf x dxf x dx=+ for any value of c.
Partial Fractions : If integrating ( ) ( ) Px Qx ∫ dx where the degree of Px( ) is smaller than the degree of Qx( ). Factor denominator as completely as possible and find the partial fraction decomposition of the rational expression. Integrate the partial fraction decomposition (P.F.D.). For each factor in the
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