Common Derivatives and Integrals Derivatives Basic Properties Formulas Rules x c is any constant n 1 n is any number c is any constant fgfgf g Product Rule fgxfgxg x Chain Rule g 2 Quotient Rule fxgxfxg x d c 0 dx cid 230 cid 246 cid 247 cid 231 ffgf g g d ln dxg x g x g x d cfxcf dx d xnx dx n d dx d dx gxg x e g x e 1 x x x x xx sin 1 sincos secsectan d dx d dx d x dx Common Derivatives Polynomials d c 0 dx Trig Functions d dx d dx Inverse Trig Functions d dx d dx Exponential Logarithm Functions d aa dx d dx Hyperbolic Trig Functions d dx d dx x sechsechtanh ln sinhcosh d dx d dx d dx d dx d dx d dx ln 1 x x 1 x sec 1 xx 0 x a 1 1 x x x x x x 1 x 2 2 d cx dx c d xnx dx n n 1 d cxncx dx n n 1 x cossin x x tansec 2 x csccsccot xx x x cotcsc 2 x 1 cos x 1 csc x 2 1 x 1 x 2 1 x 1 1 tan x 1 cot x 1 1 x 2 1 x 1 2 x e ln x x e 1 0 x x d dxx log a a x 0 1 ln x coshsinh x x tanhsech x 2 x x cschcschcoth xx cothcsch x 2 x d dx d dx d dx d dx d dx d dx Visit http tutorial math lamar edu for a complete set of Calculus I II notes 2005 Paul Dawkins Common Derivatives and Integrals Integrals c is a constant where c is a constant cid 242 b cid 242 fxgxdxfxdxgxdx a a cid 242 fxdxfxdx b cid 242 cid 242 fxgxdxfxdxgxdx Fxfxdx cid 242 bb cid 242 cid 242 aa b cid 242 a b cid 242 a 0 cdxcb a fxdxgxdx b a cid 242 fxdx b cid 242 a c c ln cid 242 cid 242 xdxxc n n n xdxxc n n 1 1 1 n 1 n 1 1 n 1 1 q q p q 1 ppp q xdxxcx qq 1 p q c 1 cid 242 2 sectanuduu c 2csccotuduu c cid 242 cid 242 uuduu c cotlnsinuduu sincosuduu c csccotcsc cid 242 cid 242 cid 242 c 1 uduuuuu 2 1 uduuuuu 2 3 c secsectanlnsectan 3 c csccsccotlncsccot a ln c u a lnlnuduuuu c u cid 242 cid 242 e uduu cid 243 cid 244 1 ln u u u 1 e c duu c lnln 0 c ln cid 242 cid 242 cid 242 dxx xdxx 1 c cfxdxcfxdx b a Basic Properties Formulas Rules cid 242 cid 242 cfxdxcfxdx b cid 242 fxdxFxFbF a a b b cid 242 cid 242 a a a cid 242 fxdx a b bc cid 242 cid 242 cid 242 fxdxfxdxfxdx c aa 0 f x on ax b then If fxg x b a on ax b then 1 axb kdxkx dxx 1 x 1 ln c a If Common Integrals Polynomials cid 242 cid 243 cid 244 cid 243 cid 244 Trig Functions cid 242 cid 242 cid 242 cid 242 cid 242 Exponential Logarithm Functions cid 242 e cid 242 cid 242 e au buduabubbu b 2 e au buduabubbu 2 cossinuduu c sectansec tanlnsecuduu coscossin uduuu csclncsccot uuduu c seclnsectan sinsincos uduuu dxaxb cid 242 cid 242 c a a c adu u du c c cid 242 b e e e c c au au u u 2 2 Visit http tutorial math lamar edu for a complete set of Calculus I II notes 2005 Paul Dawkins 2 2 2 u a 1 2 a du du du uu sin u a u 1 tan cid 246 cid 247 cid 246 cid 247 cid 230 1 cid 231 1 a 2 1 aua 2 sinhcoshuduu c sechtanhsechuduu c uduu tanhlncosh Inverse Trig Functions cid 243 cid 230 1 cid 231 cid 244 cid 230 cid 243 1 cid 244 cid 231 1 sec cid 243 cid 244 a Hyperbolic Trig Functions cid 242 cid 242 cid 242 Miscellaneous cid 243 cid 244 cid 242 cid 242 cid 242 1 ln 2 u 2 u 2 ua 1 2 du auau a 2 auduauuau 22222 2 uaduuauua 22222 2 auduau 2222 c u a 1 c c 2 u Common Derivatives and Integrals uduuuu 1 sinsin 11 c 2 tantanln 1 uduuuu 11 c 2 1 2 c uduuuu 1 coscos 11 c 2 c c u a cid 246 cid 247 cid 242 cid 242 cid 242 coshsinhuduu c cid 242 cid 242 cschcothcschuduu c cid 242 1 uduu sechtansinh c cid 242 cid 242 2 sechtanhuduu c cschcothuduu c 2 c cid 243 cid 244 1 du uaau a 2 2 1 ln 2 u a c a 2 2 a 2 2 2 c 2 ln ln sin cid 246 cid 247 a cid 230 cid 231 2 c 2 uaaa u 2 cid 230 cid 231 a cid 246 cid 247 1 auuduauu 22cos 22 cid 242 Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class u Substitution fgxgxdx Given fgxgxdxfudu b cid 242 a integral Integration by Parts The standard formulas for integration by parts are b a cid 242 cid 242 cid 242 udvuvvduudvuvvdu then the substitution bg b ag a ug x b a cid 242 cid 242 cid 242 b a will convert this into the Choose u and dv and then compute du by differentiating u and compute v by using the fact that vdv cid 242 Visit http tutorial math lamar edu for a complete set of Calculus I II notes 2005 Paul Dawkins Common Derivatives and Integrals a b a b a b Trig Substitutions If the integral contains the following root use the given substitution and formula abx 2222 cid 222 2 x sinandcos1sin qq q bxa 2222 cid 222 2 x secandtansec qq q 1 2 x tanandsec1tan qq q abx 2222 cid 222 Partial Fractions P x Q x cid 243 cid 244 If integrating dx where the degree largest exponent of P x is smaller than the Q x then factor the denominator as completely as possible and find the partial degree of 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 2axbx c A ax b Ax B axbx c 2 ax b k axbx 2 c k A 1 ax b Ax B 1 1 axbx c 2 A 2 axbax b 2 L L A k Ax B k k axbx c 2 k k Products and some Quotients of Trig Functions cid …