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DERIVATI¥ES AND INlrEGRAlS Basic Differentiation Rules I. ; r[cu] 4. i_ :::::l [!!.] = vu' v-~ uv' d S. dr [c] = 0 dr l' d 7. dx[x] == I 8. :r[I11 IJ = 10. !!.[e"] = euu' dr 13. ! [sin u] 16. = (cos u)u' :~[cot u] = - (csc 2 u)u' u' r ...;1-ud -u' 22. -d [arccot u] = - --~ r 1 + ud 19. -d [arcsin 11] = 25. : r [sinh 11] k+ u- d 11. -d [log" u] r = -(I n- a)11 14. :r[cos 11] =- d 17. dr[sec u] = (sec 11 tan u)u' ll' r d 23. d - [arcsec u] r 15. :r[tan u] (sin u)u' d 29. dr [sech u] d - u' d 1 r d ,, 34. - [coth- 1 11] = - - , dx 1 - u- d 35. dx [sech- 1 u] 21. d - [arctan u] u' = -1-+u-, 24. dx[arccsc 11] = d r d ~ ...; 1 - u- u' = Iu I...;~ u- - 1 27. : r [tanh u] = - (sech 11 tanh u)u' 32. d - [cosh- 1 u] = (sec2 11)11' 18. dx[csc 11] ""' - (esc u cot u)11' 26. :r [cosh u] = (sinh u)11' 28. : r [coth u] = - (csch2 u)u' = d ,, 9. d - [In 11] ""' r u d 12. dx[a"] = (In a)a"ll' *0 u :: (11 '), 1 1 d 20. -d [arccos u] = ~ = (cosh u)11' 31. dxd [sinh- 1 u] d 3. - [uv] = uv' + vu' dr d 6. dr[u"] = nll" - 1u' 2. ![11 ± v] ""' 11' ± v' cu' = (sech 2 u)u' d 30. dx [csch 11] "" - (csch u coth 11 )u ' d II' - u' I I ~ II ...; u- - I ,, 33. - [tanh- 1 u] = - -, dr 1 - ud - u' 36. -d [csch- 1 u] = I I Jf+lj2 r 11 I + u- =~ ~~- - I -11' = u~ 1 - 11· Basic Integration Formulas 3. J J 5. Je" d11 ., e" + C 1. 7. 9. 11. kj(11) du = k fj(11) du du = u 2. J [/(11) +C J J J 4. fa"du = Cnla)a" 6. Jsin cos u du = sin 11 + C cot u du = !n isin ul + C esc u du = - Jn lcsc 11 13. Jcsc-2 11 d11 =- cot 11 + f © Brooks/Cole, Cengage Learning 10. + cot ul + C C 12. 14. + I u , = - arctan - + C al + u· a a d11 11 du =- = Jf(u) du ± J g(11) du +C cos 11 +C 8. fran 11 du = - In Ieos 11l 15. Jcsc u cot ud11 = - esc u 17. ± g(u)] du C 16. 18. J J J sec 11 d11 = lnlsec 11 + tan u l + C sec 2 11 d11 = tan 11 +C sec u tan 11 d11 = sec u f f +C + C d11 . 11 .j , , == arcsm - + C a·- ua d11 I lui .j ~ , = - arcsec- + C 11 u- - a· a a TrRIG0NOMETRY Definition of the Six Trigonometric Functions Right triangle definitions, where 0 < 0 < Tr/2. sin 0 = opp hyp esc 0 = hyp opp d' (_!2' :!l.) (_:/l 2 . 2 !) (_il 2 •2 h cos 0 = ~ sec 0 = yp hyp adj d' tan (} = opp cot (} = ~ adj opp Circular function definitions, where 0 is any angle. .1' r= sin(} = ../.t2+ y2 --~-<o_._•> (4. 4) (4.¥) (4. ~) (-1.0) g esc(} =!:.. r il) 2 -1)2 (_il 2 _:/l.) (_:!l 2 • 2 )' t X cos(} = r X sec(} = !:. X X )' Reciprocal Identities ' .r = -1sm esc x 1 csc .r = - .sm .r sec .t = - cos .r cosx 1 =sec x Double -Angle Formulas 1 tanx "" - cotx 1 cotx = - tan .r Tangent and Cotangent Identities sin x cos .r tanx =cosx cot .r = - .smx I + cot2 x = csc2 x 1 + tan2 x = sec 2 x Power-Reducing Formulas I - cos 2u sio2 u = - - - - 2 1 + cos 2u cos~ 11 = 2 ~ I - cos 2u tan- u = - - - 1 + cos 211 Sum-to-Product Formulas Cofunction Identities x) = cos x cos(f- x) = sinx r) = cot .r csc(f- x) = secx tan(!!. 2 - ' sec(f- = sin 2u 2 sin u cos 11 cos 2u = cos2 11 - sin 2 11 = 2 cos2 u - I = I - 2 sin2 u 2 tan u tan 2u = ~ I - tan~ u ~ Pythagorean Identities sin 2 x + cos2 .t = 1 sin(f - (_!2' _il) 2 tan(} =- cot(} = ~ )' (II v) cos ("- -2- v) . u - sm . v = .,- cos("- +- \') sm . (II- -- ") s•n 2 2 . u + sm . v = 2 sm . - +sm 2 x) =esc x cot(f - x) = tan x Reduction Formulas = sin(-x) -sinx csc(-x) = -cscx sec(-x) = secx cos(-x) = cosx tan(-x) = -tanx cot(-x) = - cot.r Sum and Difference Formulas sin(u ± v) = sin u cos v ± cos u sin v cos(u ± l') = cos u cos v + sin 11 sin v tan 11 ± tan v tan (u ± v) = .....;,;;;.;.;,;.;.-=-=--1 + tan u tan '' Product-to-Sum Formulas sin 11 sin v = k[cos(rt - v) - cos(u + v)] 1 cos 11 cos v "" 2[cos(u - v) + cos(11 + v)] sin u cos v = ~[sin(u + v) + sin(u - v)] cos 11 sin v ::::: ~[sin(u + v) - sin(11 - v)] © Brooks/Cole, Cengage Learning