Common Derivatives and Integrals Derivatives Basic Properties/Formulas/Rules d ′ (cf( x)) = cf′( x) , c is any constant. fxgx( ) ±=±( ) fxgx′′( ) ( ) dx ( ) d d ( xnn) = nx −1 , n is any number. (c) = 0 , c is any constant. dx dx ′ ′ f f′′ g− fg ( fg) = f′′ g + fg – (Product Rule) = – (Quotient Rule) gg2 d ( f( gx( ))) = f′′( gx( )) g( x) (Chain Rule) dx d gx( ) gx( ) d gx′( ) ee= gx′( ) (ln gx( )) = dx ( ) dx g( x) Common Derivatives Polynomials d d d d d (c) = 0 ( x) =1 (cx) = c ( xnn) = nx −1 (cxnn) = ncx −1 dx dx dx dx dx Trig Functions d d d (sinxx) = cos (cosxx) = − sin (tanxx) = sec2 dx dx dx d d d (secx) = sec xx tan (cscx) = − csc xx cot (cotxx) = − csc2 dx dx dx Inverse Trig Functions d 1 d 1 d 1 −1 = −1 = − −1 = (sin x) (cos x) (tan x) 2 dx 1− x2 dx 1− x2 dx 1+ x d 1 d 1 d 1 −1 = −1 = − −1 = − (sec x) (csc x) (cot x) 2 dx xx2 −1 dx xx2 −1 dx 1+ x Exponential/Logarithm Functions d d (aaaxx) = ln ( ) (eexx) = dx dx d 1 d 1 d 1 (ln( xx)) = , > 0 (lnxx) = , ≠ 0 (log ( xx)) =, > 0 dx x dx x dx a xln a Hyperbolic Trig Functions d d d (sinhxx) = cosh (coshxx) = sinh (tanhxx) = sech 2 dx dx dx d d d (sechx) = − sech xx tanh (cschx) = − csch xx coth (cothxx) = − csch 2 dx dx dx Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Common Derivatives and Integrals Integrals Basic Properties/Formulas/Rules ∫∫cf( x) dx= c f( x) dx , c is a constant. ∫f( x) ±= g( x) dx ∫∫ f( x) dx ± g( x) dx b b f( x) dx= F( x) = F( b) − F( a) where F( x) = f( x) dx ∫a a ∫ bb b bb cf( x) dx= c f( x) dx , c is a constant. f( x) ±= g( x) dx f( x) dx ± g( x) dx ∫∫aa ∫a ∫∫ aa a ba f( x) dx = 0 f( x) dx= − f( x) dx ∫a ∫∫ab b cb b f( x) dx= f( x) dx + f( x) dx c dx= c( b − a) ∫∫∫a ac ∫a b If fx( ) ≥ 0 on axb≤≤ then f( x) dx ≥ 0 ∫a bb If f( x) ≥ gx( ) on axb≤≤ then f( x) dx≥ g( x) dx ∫∫aa Common Integrals Polynomials 1 dx= x + c k dx= k x + c xnn dx= x+1 + c,1 n ≠− ∫ ∫ ∫ n +1 ⌠ 1 −1 −nn1 −+1 dx=ln x + c x dx=ln x + c x dx= x +≠ c,1 n ⌡ x ∫ ∫ −+n 1 p p pq+ +1 ⌠ 11 qq1 q q dx=ln ax ++ b c x dx= x += c x + c ∫ p ⌡ ax+ b a q ++1 pq Trig Functions ∫ cosu du= sin u + c ∫sinu du=−+ cos u c ∫sec2 u du= tan u + c ∫secu tan u du= sec u + c ∫ cscu cot udu=−+ csc u c ∫ csc2 u du=−+ cot u c ∫ tanu du= ln sec u + c ∫ cotu du= ln sin u + c 1 secu du= ln sec u ++ tan u c sec3 u du=( sec u tan u + ln sec u ++ tan u) c ∫ ∫ 2 1 cscu du= ln csc u −+ cot u c csc3 u du=−( csc u cot u + ln csc u −+ cot u) c ∫ ∫ 2 Exponential/Logarithm Functions au eeuudu= + c au du= + c lnu du= u ln ( u) −+ u c ∫ ∫ ln a ∫ eau eau sin (bu) du=( asin( bu) −+ b cos( bu)) c ueeuu du=−+( u1) c ∫ ab22+ ∫ au au e ⌠ 1 e cos(bu) du=( acos( bu) ++ b sin ( bu)) c du=ln ln u + c ∫ ab22+ ⌡ uuln Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Common Derivatives and Integrals Inverse Trig Functions ⌠ 1 −1 u −−1 12 du =sin + c ∫sinu du= u sin u +−+ 1 u c ⌡ au22− a ⌠ 11−1 u −−111 2 du =tan + c tanu du= u tan u − ln( 1 ++ u) c ⌡ au22+ a a ∫ 2 ⌠ 11−1 u −−1 12 du =sec + c ∫ cosu du= u cos u −−+ 1 u c ⌡ uu22− a aa Hyperbolic Trig Functions ∫sinhu du= cosh u + c ∫sechu tanh u du=−+ sech u c ∫sech2 u du= tanh u + c ∫ coshu du= sinh u + c ∫ cschu coth u du=−+ csch u c ∫ csch2 u du=−+ coth u c ∫ tanhu du= ln( cosh u) + c ∫sechu du = tan−1 sinh u+ c Miscellaneous ⌠ 11ua+ ⌠ 11ua− du = ln + c du = ln + c ⌡ a22−− u2 a ua ⌡ u22−+ a2 a ua ua2 auduau22+ = 22 ++ln u + au22 + + c ∫ 22 ua2 uaduua22− = 22 −−ln u + ua22 − + c ∫ 22 2 22u 22 au− 1 auduau− = −+sin +c ∫ 22a 2 2ua−−21 a− au 2au− u du =2 au −+ u cos +c ∫ 22a Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class. u Substitution b Given f g( x) g′( x) dx then the substitution u= gx( ) will convert this into the ∫a ( ) b gb( ) integral, f( g( x)) g′( x) dx= f( u) du . ∫∫a ga( ) Integration by Parts The standard formulas for integration by parts are, bbb udv= uv−= vdu udv uv− vdu ∫∫ ∫aaa ∫ Choose u and dv and then compute du by differentiating u and compute v by using the fact that v= ∫ dv . Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Common Derivatives and Integrals Trig Substitutions If the integral contains the following root use the given substitution and formula. a a2− bx 22 ⇒= x sinθ and cos2 θθ=− 1sin 2 b a bx22−⇒= a 2 x secθ and tan2 θθ=− sec 2 1 b a a2+ bx 22 ⇒= x tanθ and sec2 θθ=+ 1 tan 2 b Partial Fractions ⌠ Px( ) If integrating dx where the degree (largest exponent) of Px( ) is smaller than the ⌡ Qx( ) degree of Qx( ) then factor the 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 denominator we get term(s) in the decomposition according to the following table. Factor in Qx( ) Term in P.F.D Factor in Qx( ) Term in P.F.D AA A A k 12+ ++ k ax+ b (ax+ b) 2 k ax+ b ax+ b (ax++ b) ( ax b) Ax11+ B Axkk+ B Ax+ B 2 k ++ 2 2 k ax++ bx c 2 (ax++ bx c) 2 ax++ bx c ax++ bx c (ax++ bx c) Products and (some) Quotients of Trig Functions ∫sinnmx cos x dx 1. If n is odd. Strip one sine out and convert the remaining sines to cosines using sin22xx= 1 − cos , then use the substitution ux= cos 2. If m is odd. Strip one cosine out and convert the remaining cosines to sines using cos22xx= 1 − sin , then use the substitution ux= sin 3. If n and m are both odd. Use either 1. or 2. 4. If n and m are both even. Use double angle formula for sine and/or half angle formulas to reduce the integral into a form that can be integrated. ∫ tannmx sec x dx 1. If n is odd. Strip one tangent and one secant out and convert the remaining 22 tangents to secants using tanxx= sec − 1, then use the substitution ux= sec 2. If m is even. Strip two secants out and convert the remaining secants to tangents 22 using secxx= 1 + tan , then use the substitution ux= tan 3. If n is odd and m is even. Use either 1. or 2. 4. If n is even and m is odd. Each integral will be dealt with differently. 33 Convert Example : cos62xx=( cos) =( 1 − sin 2 x) Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins .