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 ()( )()dcfxcfxdx¢= , c is any constant. () ()( )() ()fxgxfxgx¢¢¢±=± ()1nndxnxdx-= , n is any number. ()0dcdx=, c is any constant. ( )fgfgfg¢¢¢=+ – (Product Rule) 2ffgfggg¢¢¢æö-=ç÷èø – (Quotient Rule) ()( )( )()( )()dfgxfgxgxdx¢¢= (Chain Rule) ( )()()()gxgxdgxdx¢=ee ()( )()()lngxdgxdxgx¢= Common Derivatives Polynomials ()0dcdx= ()1dxdx= ( )dcxcdx= ( )1nndxnxdx-= ( )1nndcxncxdx-= Trig Functions ( )sincosdxxdx= ( )cossindxxdx=- ( )2tansecdxxdx= ( )secsectandxxxdx= ( )csccsccotdxxxdx=- ( )2cotcscdxxdx=- Inverse Trig Functions ( )121sin1dxdxx-=- ( )121cos1dxdxx-=-- ( )121tan1dxdxx-=+ ()121sec1dxdxxx-=- ( )121csc1dxdxxx-=-- ( )121cot1dxdxx-=-+ Exponential/Logarithm Functions ( )()lnxxdaaadx= ( )xxddx=ee ()( )1ln,0dxxdxx=> ( )1ln,0dxxdxx=¹ ()( )1log,0lnadxxdxxa=> Hyperbolic Trig Functions ( )sinhcoshdxxdx= ( )coshsinhdxxdx= ( )2tanhsechdxxdx= ( )sechsechtanhdxxxdx=- ( )cschcschcothdxxxdx=- ( )2cothcschdxxdx=- 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 ()()cfxdxcfxdx=òò, c is a constant. ()()()()fxgxdxfxdxgxdx±=±òòò () () () ()bbaafxdxFxFbFa==-ò where ()()Fxfxdx=ò () ()bbaacfxdxcfxdx=òò, c is a constant. () () () ()bbbaaafxgxdxfxdxgxdx±=±òòò ()0aafxdx=ò () ()baabfxdxfxdx=-òò () () ()bcbaacfxdxfxdxfxdx=+òòò ()bacdxcba=-ò If ()0fx³ on axb££ then ()0bafxdx³ò If ()()fxgx³ on axb££ then () ()bbaafxdxgxdx³òò Common Integrals Polynomials dxxc=+ò kdxkxc=+ò 11,11nnxdxxcnn+=+¹-+ò 1lndxxcx=+óôõ 1lnxdxxc-=+ò 11,11nnxdxxcnn--+=+¹-+ò 11lndxaxbcaxba=+++óôõ 111pppqqqqpqqxdxxcxcpq++=+=+++ò Trig Functions cossinuduuc=+ò sincosuduuc=-+ò 2sectanuduuc=+ò sectansecuuduuc=+ò csccotcscuuduuc=-+ò 2csccotuduuc=-+ò tanlnsecuduuc=+ò cotlnsinuduuc=+ò seclnsectanuduuuc=++ò ( )31secsectanlnsectan2uduuuuuc=+++ò csclncsccotuduuuc=-+ò ( )31csccsccotlncsccot2uduuuuuc=-+-+ò Exponential/Logarithm Functions uuduc=+òee lnuuaaduca=+ò ()lnlnuduuuuc=-+ò ( ) ( ) ( )( )22sinsincosauaubuduabubbucab=-++òee ()1uuuduuc=-+òee () ()()( )22coscossinauaubuduabubbucab=+++òee 1lnlnlnduucuu=+óôõCommon Derivatives and Integrals Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Inverse Trig Functions 1221sinuducaau-æö=+ç÷èø-óôõ 112sinsin1uduuuuc--=+-+ò 12211tanuducauaa-æö=+ç÷+èøóôõ ( )1121tantanln12uduuuuc--=-++ò 12211secuducaauua-æö=+ç÷èø-óôõ 112coscos1uduuuuc--=--+ò Hyperbolic Trig Functions sinhcoshuduuc=+ò coshsinhuduuc=+ò 2sechtanhuduuc=+ò sechtanhsechuduuc=-+ò cschcothcschuduuc=-+ò 2cschcothuduuc=-+ò ()tanhlncoshuduuc=+ò 1sechtansinhuduuc-=+ò Miscellaneous 2211ln2uaducauaua+=+--óôõ 2211ln2uaducuaaua-=+-+óôõ 2222222ln22uaauduauuauc+=+++++ò 2222222ln22uauaduuauuac-=--+-+ò 222221sin22uauauduauca-æö-=-++ç÷èøò 222122cos22uaaauauuduauuca---æö-=-++ç÷èøò Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class. u Substitution Given ()( )()bafgxgxdx¢ò then the substitution ()ugx= will convert this into the integral, ()( )() ()()()bgbagafgxgxdxfudu¢=òò. Integration by Parts The standard formulas for integration by parts are, bbbaaaudvuvvduudvuvvdu=-=-òòòò Choose u and dv and then compute du by differentiating u and compute v by using the fact that vdv=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. 22222sinandcos1sinaabxxbqqq-Þ==- 22222secandtansec1abxaxbqqq-Þ==- 22222tanandsec1tanaabxxbqqq+Þ==+ Partial Fractions If integrating ()()PxdxQxóôõ where the degree (largest exponent) of ()Px is smaller than the 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 axb+ Aaxb+ ( )kaxb+ ( ) ( )122kkAAAaxbaxbaxb++++++L 2axbxc++ 2AxBaxbxc+++ ()2kaxbxc++ ( )1122kkkAxBAxBaxbxcaxbxc++++++++L Products and (some) Quotients of Trig Functions sincosnmxxdxò 1. If n is odd. Strip one sine out and convert the remaining sines to cosines using 22sin1cosxx=- , then use the substitution cosux= 2. If m is odd. Strip one cosine out and convert the remaining cosines to sines using 22cos1sinxx=- , then use the substitution sinux= 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. tansecnmxxdxò 1. If n is odd. Strip one tangent and one secant out and convert the remaining tangents to secants using 22tansec1xx=-, then use the substitution secux= 2. If m is even. Strip two secants out and convert the remaining secants to tangents using 22sec1tanxx=+ , then use the substitution tanux= 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 :