Purdue STAT 51100 - Lecture notes (2 pages)

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Lecture notes

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Lecture notes

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2
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Purdue University
Course:
Stat 51100 - Statistical Methods
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Common Derivatives and Integrals Common Derivatives and Integrals Derivatives Integrals Basic Properties Formulas Rules d cf x cf x c is any constant f x g x f x g x dx d n d c 0 c is any constant x nxn 1 n is any number dx dx f f g f g Quotient Rule f g f g f g Product Rule g2 g d f g x f g x g x Chain Rule dx g x d d g x g x ln g x e g x e dx g x dx Common Derivatives Polynomials d d c 0 x 1 dx dx d cx c dx Trig Functions d sin x cos x dx d sec x sec x tan x dx d cos x sin x dx d csc x csc x cot x dx d tan x sec2 x dx d cot x csc2 x dx Inverse Trig Functions d 1 sin 1 x dx 1 x2 d sec 1 x 12 dx x x 1 d 1 cos 1 x dx 1 x2 d csc 1 x 12 dx x x 1 d 1 tan 1 x dx 1 x2 d cot 1 x 1 1x2 dx d n x nxn 1 dx d cxn ncxn 1 dx 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 a f x dx F x a F b F a where F x f x dx b b b b b b b a cf x dx c a f x dx c is a constant a f x g x dx a f x dx a g x dx a b a f x dx 0 b a a f x dx b f x dx c b b a f x dx a f x dx c f x dx If f x 0 on a x b then a c dx c b a b a f x dx 0 If f x g x on a x b then b b a f x dx a g x dx Common Integrals Polynomials 1 dx x c k dx k x c x dx n 1 x 1 dx ln x c x x x 1 n dx ln x c 1 dx 1 ln ax b c ax b a x p q n dx p q c n 1 1 x n 1 c n 1 n 1 p dx n 1 1 q 1 q x c x p q 1 Trig Functions cos u du sin u c p q q c sin u du cos u c sec u du tan u c sec u tan u du sec u c csc u cot udu csc u c csc u du cot u c tan u du ln sec u c cot u du ln sin u c 1 sec u du ln sec u tan u c sec u du 2 sec u tan u ln sec u tan u c 2 2 3 Exponential Logarithm Functions d x d x a a x ln a e ex dx dx d d ln x 1x x 0 ln x 1x x 0 dx dx Hyperbolic Trig Functions d sinh x cosh x dx d sech x sech x tanh x dx csc u du ln csc u cot u c d log a x x ln1 a x 0 dx Visit http tutorial math lamar edu for a complete set of Calculus I II notes 2005 Paul 3 u du 1 csc u cot u ln csc u cot u c 2 Exponential Logarithm Functions u u e du e c d d cosh x sinh x tanh x sech 2 x dx dx d d csch x csch x coth x coth x csch 2 x dx dx csc u a du au c ln a e au a sin bu b cos bu c a b2 e au au e cos bu du a cos bu b sin bu c a 2 b2 e Dawkins au sin bu du 2 Visit http tutorial math lamar edu for a complete set of Calculus I II notes ln u du u ln u u c ue du u 1 e u u c 1 du ln ln u c u ln u 2005 Paul Dawkins Common Derivatives and Integrals Inverse Trig Functions 1 u du sin 1 c 2 2 a a u sin 1 1 u du tan 1 c 2 2 a a u a 1 1 u du sec 1 c a a u u2 a2 Hyperbolic Trig Functions sinh u du cosh u c sech tanh u du sech u c tanh u du ln cosh u c 1 tan u du u sin 1 u 1 u 2 c 1 u du u tan 1 u ln 1 u 2 c 2 1 cos 1 u du u cos 1 u 1 u 2 c cosh u du sinh u c sech csch coth u du csch u c csch sech u du tan sinh u c Miscellaneous 1 du 1 ln u a c 2 a u2 2a u a 2au u 2 du u du tanh u c 2 u du coth u c 1 du 1 ln u a c 2 u a2 2a u a 2 1 u a2 a u du a 2 u 2 ln u a 2 u 2 c 2 2 2 u a u 2 a 2 du u 2 a 2 ln u u 2 a 2 c 2 2 u 2 a2 u 2 2 2 a u du a u sin 1 c 2 2 a 2 Common Derivatives and Integrals Factor in Q x 2 u Substitution a f g x g x dx then the substitution u g x will convert this into the b g b integral f g x g x dx f u du a g a b Integration by Parts The standard formulas for integration by parts are udv uv vdu b b b a udv uv a a vdu 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 Term in P F D Factor in Q x ax b A ax b ax 2 bx c Ax B ax 2 bx c ax b ax 2 Term in P F D A1 A2 Ak L k ax b ax b 2 ax b k bx c Ak x Bk A1 x B1 L k 2 ax bx c ax2 bx c k Products and some Quotients of Trig Functions n m sin x cos x dx u a a2 a u 2 au u 2 cos 1 c 2 2 a Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class Given Trig Substitutions …

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