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

∫ ∫cf x dx c f x dx( ) = ( ), is a constantc bb ( ) ( ) aa ∫∫cf x dx c f x dx= , is a constantc b ( ) a ∫cdx c b a= − bb ( ) ( ) aa ∫∫f x dx f x dx≤ b cb ( ) ( ) ( ) a ac ∫ ∫∫f x dx f x dx f x dx= + for any value of . c If f x gx( )≥ ( ) ona xb≤≤ then ( ) ( ) b b a a ∫∫f x dx g x dx≥ Iff x( )≥0 ona xb≤≤ ...

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