What is Jacobian? | The right way of thinking derivatives and integrals
What is Jacobian? | The right way of thinking derivatives and integrals

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Common derivatives integrals
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Apr. 26, 2015

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Common derivatives integrals
Apr. 26, 2015

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olziich
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Common derivatives integrals
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Common derivatives integrals
Common Derivatives and
Integrals Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Derivatives Basic Properties/Formulas/Rules ( )( ) ( ) d cf x cf x dx ¢= , c is any constant. ( ) ( )( ) ( ) ( )f x g x f x g x¢ ¢ ¢± = ± ( ) 1n nd x nx dx – = , n is any number. ( ) 0 d c dx = , c is any constant. ( )f g f g f g¢ ¢ ¢= + – (Product Rule) 2 f f g f g g g ¢ ¢ ¢æ ö – =ç ÷ è ø – (Quotient Rule) ( )( )( ) ( )( ) ( ) d f g x f g x g x dx ¢ ¢= (Chain Rule) ( ) ( ) ( ) ( )g x g xd g x dx ¢=e e ( )( ) ( ) ( ) ln g xd g x dx g x ¢ = Common Derivatives Polynomials ( ) 0 d c dx = ( ) 1 d x dx = ( ) d cx c dx = ( ) 1n nd x nx dx – = ( ) 1n nd cx ncx dx – = Trig Functions ( )sin cos d x x dx = ( )cos sin d x x dx = – ( ) 2 tan sec d x x dx = ( )sec sec tan d x x x dx = ( )csc csc cot d x x x dx = – ( ) 2 cot csc d x x dx = – Inverse Trig Functions ( )1 2 1 sin 1 d x dx x – = – ( )1 2 1 cos 1 d x dx x – = – – ( )1 2 1 tan 1 d x dx x – = + ( )1 2 1 sec 1 d x dx x x – = – ( )1 2 1 csc 1 d x dx x x – = – – ( )1 2 1 cot 1 d x dx x – = – + Exponential/Logarithm Functions ( ) ( )lnx xd a a a dx = ( )x xd dx =e e ( )( ) 1 ln , 0 d x x dx x = > ( ) 1 ln , 0 d x x dx x = ¹ ( )( ) 1 log , 0 ln a d x x dx x a = > Hyperbolic Trig Functions ( )sinh cosh d x x dx = ( )cosh sinh d x x dx = ( ) 2 tanh sech d x x dx = ( )sech sech tanh d x x x dx = – ( )csch csch coth d x x x dx = – ( ) 2 coth csch d x x dx = –
Common Derivatives and
Integrals Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Integrals Basic Properties/Formulas/Rules ( ) ( )cf x dx c f x dx=ò ò , c is a constant. ( ) ( ) ( ) ( )f x g x dx f x dx g x dx± = ±ò ò ò ( ) ( ) ( ) ( ) b b aa f x dx F x F b F a= = -ò where ( ) ( )F x f x dx= ò ( ) ( ) b b a a cf x dx c f x dx=ò ò , c is a constant. ( ) ( ) ( ) ( ) b b b a a a f x g x dx f x dx g x dx± = ±ò ò ò ( ) 0 a a f x dx =ò ( ) ( ) b a a b f x dx f x dx= -ò ò ( ) ( ) ( ) b c b a a c f x dx f x dx f x dx= +ò ò ò ( ) b a cdx c b a= -ò If ( ) 0f x ³ on a x b£ £ then ( ) 0 b a f x dx ³ò If ( ) ( )f x g x³ on a x b£ £ then ( ) ( ) b b a a f x dx g x dx³ò ò Common Integrals Polynomials dx x c= +ò k dx k x c= +ò 11 , 1 1 n n x dx x c n n + = + ¹ – +ò 1 lndx x c x = +ó ô õ 1 lnx dx x c- = +ò 11 , 1 1 n n x dx x c n n – – + = + ¹ – +ò 1 1 lndx ax b c ax b a = + + + ó ô õ 11 1 p p p q q q q p q q x dx x c x c p q + + = + = + + +ò Trig Functions cos sinu du u c= +ò sin cosu du u c= – +ò 2 sec tanu du u c= +ò sec tan secu u du u c= +ò csc cot cscu udu u c= – +ò 2 csc cotu du u c= – +ò tan ln secu du u c= +ò cot ln sinu du u c= +ò sec ln sec tanu du u u c= + +ò ( )3 1 sec sec tan ln sec tan 2 u du u u u u c= + + +ò csc ln csc cotu du u u c= – +ò ( )3 1 csc csc cot ln csc cot 2 u du u u u u c= – + – +ò Exponential/Logarithm Functions u u du c= +òe e ln u u a a du c a = +ò ( )ln lnu du u u u c= – +ò ( ) ( ) ( )( )2 2 sin sin cos au au bu du a bu b bu c a b = – + +ò e e ( )1u u u du u c= – +ò e e ( ) ( ) ( )( )2 2 cos cos sin au au bu du a bu b bu c a b = + + +ò e e 1 ln ln ln du u c u u = +ó ô õ
Common Derivatives and
Integrals Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Inverse Trig Functions 1 2 2 1 sin u du c aa u – æ ö = +ç ÷ è ø- ó ô õ 1 1 2 sin sin 1u du u u u c- – = + – +ò 1 2 2 1 1 tan u du c a u a a – æ ö = +ç ÷ + è ø ó ô õ ( )1 1 21 tan tan ln 1 2 u du u u u c- – = – + +ò 1 2 2 1 1 sec u du c a au u a – æ ö = +ç ÷ è ø- ó ô õ 1 1 2 cos cos 1u du u u u c- – = – – +ò Hyperbolic Trig Functions sinh coshu du u c= +ò cosh sinhu du u c= +ò 2 sech tanhu du u c= +ò sech tanh sechu du u c= – +ò cschcoth cschu du u c= – +ò 2 csch cothu du u c= – +ò ( )tanh ln coshu du u c= +ò 1 sech tan sinhu du u c- = +ò Miscellaneous 2 2 1 1 ln 2 u a du c a u a u a + = + – – ó ô õ 2 2 1 1 ln 2 u a du c u a a u a – = + – + ó ô õ 2 2 2 2 2 2 2 ln 2 2 u a a u du a u u a u c+ = + + + + +ò 2 2 2 2 2 2 2 ln 2 2 u a u a du u a u u a c- = – – + – +ò 2 2 2 2 2 1 sin 2 2 u a u a u du a u c a – æ ö – = – + +ç ÷ è ø ò 2 2 2 1 2 2 cos 2 2 u a a a u au u du au u c a — -æ ö – = – + +ç ÷ è ø ò Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class. u Substitution Given ( )( ) ( ) b a f g x g x dx¢ò then the substitution ( )u g x= will convert this into the integral, ( )( ) ( ) ( )( ) ( )b g b a g a f g x g x dx f u du¢ =ò ò . Integration by Parts The standard formulas for integration by parts are, b bb aa a udv uv vdu udv uv vdu= – = -ò ò ò ò Choose u and dv and then compute du by differentiating u and compute v by using the fact that v dv= ò .
Common Derivatives and
Integrals Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Trig Substitutions If the integral contains the following root use the given substitution and formula. 2 2 2 2 2 sin and cos 1 sin a a b x x b q q q- Þ = = – 2 2 2 2 2 sec and tan sec 1 a b x a x b q q q- Þ = = – 2 2 2 2 2 tan and sec 1 tan a a b x x b q q q+ Þ = = + Partial Fractions If integrating ( ) ( ) P x dx Q x ó ô õ where the degree (largest exponent) of ( )P x is smaller than the degree of ( )Q x 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 ( )Q x Term in P.F.D Factor in ( )Q x Term in P.F.D ax b+ A ax b+ ( ) k ax b+ ( ) ( ) 1 2 2 k k AA A ax b ax b ax b + + + + + + L 2 ax bx c+ + 2 Ax B ax bx c + + + ( )2 k ax bx c+ + ( ) 1 1 2 2 k k k A x BA x B ax bx c ax bx c ++ + + + + + + L Products and (some) Quotients of Trig Functions sin cosn m x xdxò 1. If n is odd. Strip one sine out and convert the remaining sines to cosines using 2 2 sin 1 cosx x= – , then use the substitution cosu x= 2. If m is odd. Strip one cosine out and convert the remaining cosines to sines using 2 2 cos 1 sinx x= – , then use the substitution sinu x= 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. tan secn m x x dxò 1. If n is odd. Strip one tangent and one secant out and convert the remaining tangents to secants using 2 2 tan sec 1x x= – , then use the substitution secu x= 2. If m is even. Strip two secants out and convert the remaining secants to tangents using 2 2 sec 1 tanx x= + , then use the substitution tanu x= 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. Convert Example : ( ) ( ) 3 36 2 2 cos cos 1 sinx x x= = –

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