Common Derivatives Integrals - cheat sheets

1 Common Derivatives and Integrals Visit 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 -= (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= ()()1log,0lnadxxdxxa=> Hyperbolic Trig Functions ()sinhcoshdxxdx= ()coshsinhdxxdx= ()2tanhsechdxxdx= ()sechsechtanhdxxxdx=- ()cschcschcothdxxxdx=- ()2cothcschdxxdx=- Common Derivatives and Integrals Visit for a complete set of Calculus I & II notes.

2 2005 Paul Dawkins Integrals Basic Properties/Formulas/Rules ()()cfxdxcfxdx= , c is a constant. ()()()()fxgxdxfxdxgxdx = ()()()()bbaafxdxFxFbFa==- where ()()Fxfxdx= ()()bbaacfxdxcfxdx= , c is a constant. ()()()()bbbaaafxgxdxfxdxgxdx = ()0aafxdx= ()()baabfxdxfxdx=- ()()()bcbaacfxdxfxdxfxdx=+ ()bacdxcba=- If ()0fx on axb then ()0bafxdx If ()()fxgx on axb then ()()bbaafxdxgxdx Common Integrals Polynomials dxxc=+ kdxkxc=+ 11,11nnxdxxcnn+=+ -+ 1lndxxcx=+ 1lnxdxxc-=+ 11,11nnxdxxcnn--+=+ -+ 11lndxaxbcaxba=+++ 111pppqqqqpqqxdxxcxcpq++=+=+++ Trig Functions cossinuduuc=+ sincosuduuc=-+ 2sectanuduuc=+ sectansecuuduuc=+ csccotcscuuduuc=-+ 2csccotuduuc=-+ tanlnsecuduuc=+ cotlnsinuduuc=+ seclnsectanuduuuc=++ ()31secsectanlnsectan2uduuuuuc=+++ csclncsccotuduuuc=-+ ()31csccsccotlncsccot2uduuuuuc=-+-+ Exponential/Logarithm Functions uuduc=+ ee lnuuaaduca=+ ()lnlnuduuuuc=-+ ()()()()22sinsincosauaubuduabubbucab=-++ ee ()1uuuduuc=-+ ee ()()()()22coscossinauaubuduabubbucab=+++ ee 1lnlnlnduucuu=+ Common Derivatives and Integrals Visit for a complete set of Calculus I & II notes.

3 2005 Paul Dawkins Inverse Trig Functions 1221sinuducaau- =+ - 112sinsin1uduuuuc--=+-+ 12211tanuducauaa- =+ + ()1121tantanln12uduuuuc--=-++ 12211secuducaauua- =+ - 112coscos1uduuuuc--=--+ Hyperbolic Trig Functions sinhcoshuduuc=+ coshsinhuduuc=+ 2sechtanhuduuc=+ sechtanhsechuduuc=-+ cschcothcschuduuc=-+ 2cschcothuduuc=-+ ()tanhlncoshuduuc=+ 1sechtansinhuduuc-=+ Miscellaneous 2211ln2uaducauaua+=+-- 2211ln2uaducuaaua-=+-+ 2222222ln22uaauduauuauc+=+++++ 2222222ln22uauaduuauuac-=--+-+ 222221sin22uauauduauca- -=-++ 222122cos22uaaauauuduauuca--- -=-++ 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 then the substitution ()ugx= will convert this into the integral, ()()()()()()bgbagafgxgxdxfudu = . Integration by Parts The standard formulas for integration by parts are, bbbaaaudvuvvduudvuvvdu=-=- Choose u and dv and then compute du by differentiating u and compute v by using the fact that vdv=.

4 Common Derivatives and Integrals Visit 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-fi==- 22222secandtansec1abxaxbqqq-fi==- 22222tanandsec1tanaabxxbqqq+fi==+ Partial Fractions If integrating ()()PxdxQx 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 ( ). For each factor in the denominator we get term(s) in the decomposition according to the following table. Factor in ()Qx Term in Factor in ()Qx Term in axb+ Aaxb+ ()kaxb+ ()()122kkAAAaxbaxbaxb++++++L 2axbxc++ 2 AxBaxbxc+++ ()2kaxbxc++ ()1122kkkAxBAxBaxbxcaxbxc++++++++L Products and (some) Quotients of Trig Functions sincosnmxxdx 1. If n is odd. Strip one sine out and convert the remaining sines to cosines using 22sin1cosxx=-, then use the substitution cosux= 2.

5 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. tansecnmxxdx 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 : ()()33622coscos1sinxxx==.