Math Fomulas

PllUJPERTY OF vvcr FARGO 0099031 ,A L G E B RA ( " "; , GEOMETR , ' ' ARITHMETIC OPERATIONS GI:OI\tJETRIC FORMULAS a c ad + be a(b + c) = ab + ac -+-=-- b d bd a a + cae bad ad --=-+-. -=-x-= b b b c b c be d EXPONENTS AND RADICALS m X 7=x /11-1/ (x"')" = x TnIl X-II x" (xy)" = x"y" 'y~=%=m'$ i;=~ FACTORING SPECIAL POLYNOMIALS 2 x - i = (x + y) (x - y) 2 x 3 + / = (x + y) (x - xy + y') x 3 - / = (x - y) (x 2 + xy + y2) BINOMIAL THEOREM 2 (x + y)2 = x 2 + 2xy + i (x - y)2 = x - 2xy + y' (x + y)3 = x 3 + :'1x 2 y + 3xy' + / 3 (x - y)3 = x - 3x 2 y + 3xy' - / n(n - I) (x + y)" = x" + nx"-'y + X"-2y' 2 + ' , , + (~)x"-kl + ' , , + nxy"-' + y" n) _n..:....(n_-----'l)_'_'_. ..:....(n_-_k_-+_1..:....) where = ( k 1·2·3· .... k QUADRATIC FORMULA lfax 2 + bx + c = r ,- 4ac INEQUALITIES AND A VALUE If a < band b < c, then a < c. If a < b, then a + ,< b + c. Tf a .' , ca < cb. 1 < l; < 0, thL 'l > cb. ,f a> 0 means A : a or x = -a means -a <. x < a means x > a or x < -a ,Formulas for area A, circUmference C, and volume V: Triangle Circle Sector of Circle A = ibh A = 7rr 2 A = ir 2 0 = iabsinO C = 27rr s = rO (0 in radians) ~ b Sphere Cylinder V = ~7rr3 V = 7rr 2 h A = 47rr 2 T h 1 DISTANCE AND MIDPOINT FORMULAS Distance between P, (x" y,) and P2(X2, Y2): - (x, +X2 y, +Y2) Midpoint of P,P2 :. --2-'--2 LINES Slope of line through P,(x"y,) and P2(X2,Y2): Y2 - y, m=-- X2 - Xl Point-slope equation of line through P,(x"y,) with slope m: y - y, = m(x - x,) Slope-intercept equation of line with slope m and y-intercept b: y=mx+b CIRCLES Equation of the circle with center th, k) and radius r: 2 (x - h/ + (y - k)2 = r I ~£ t'4, TRIGONOMETRY ANGLE MEASUREMENT FUNDAMENTAL IDENTITIES 1 1 7T radians = 180 0 cscO =- secO =- sin 0 cosO 7T 180 0 1° = -rad 1 rad =- sinO cosO 180 . 7T tanO =- cotO =- cosO sinO s = rO 1 sin 2 0 + cos 2 0 = 1 cotO =- (0 in radians) tanO 2 0 2 0 RIGHT ANGLE TRIGONOMETRY cos( -0) = cos 0 sin(-O) = -sinO 1 + tan = sec sin 0 = opp cscO =- hyp sin(; - 0) = cosO tan(-O) = -tanO hyp opp adj secO = hyp cosO = cos(; - 0) = sinO tan(; - 0) = cotO hyp adj opp tanO =- cotO = ~ adj opp THE LAW OF SINES B sin A sin B sinC TRIGONOMETRIC FUNCTIONS a b c r y c sinO =.1'. cscO = r y x r THE LAW OF COSINES cosO =- secO = r x a 2 = b 2 + c 2 - 2bccosA x 2 2 b 2 tanO = .1'. cotO =- x = a + c - 2accosB x y A 2 2 c = a + b 2 - 2abcosC GRAPHS OF THE TRIGONOMETRIC FUNCTIONS y .., , y -I \ y y = sinx y = esc x y V: :\ I I I I Tr 2Tr X -I hi I I I i\ I y ADDITION AND SUBTRACTION FORMULAS y = cosx sin(x + y) = sin x cos y + cos x siny sin(x - y) = sin x cos y - cos x siny cos(x + y) = cosxcosy --, sinxsiny cos(x - y) = cosxcosy + sinxsiny tan x + tany I . tan(x + y) = v 1 - tanxtany y = secx tanx - tany I \/tan(x - y) = 1 + tan x tany I I I I !~ hi 2Tr x I Tr I DOUBLE-ANGLE FORMULAS I I I sin 2x = 2 sin xcosx 2 2 I.. cos 2x = cos x - sin 2 x = 2 cos x - 1 = 1 - 2 sin 2 x TRIGONOMETRIC FUNCTIONS OF IMPORTANT ANGLES 2 tanx . tan 2x = , 0 1 - tan-x radians sinO cosO tanO 0° 0 0 1 0 30 0 7T/6 1/2 ../3/2 ../3/3 HALF-ANGLE FORMULAS 45 0 7T/4 vIz/2 vIz/2 1 60° 7T/3 ../3/2 1/2 ../3 ? 2 1 - cos 2x 2 1 + cos2x sm x = cos x = --- 90° 7T/2 1 0 2 2 DIFFERENTIATION RULES GENERAL FORMULAS d d 2. - [cf(x)] = ej'(x) 1. dx (e) = 0 d, d 3. - [j(x) + g(x)] = j'(x) + g'(x) 4. !!- [j(x) - g(x)] = j'(x) - g'(x) dx dx d 6 !!- [ j(X)] = g(x)j'(x) - j(x)g'(x) 5. dx [j(x)g(x)] = j(x)g'(x) + g(x)j'(x) (Product Rule) . dx g(x) [g(x)]' (Quotient Rule) d d 7. dx j(g(x)) = j'(g(x))g'(x) (Chain Rule) 8. - (x") = IlX,,-1 (Power Rule) dx EXPONENTIAL AND LOGARITHMIC FUNCTIONS d d 9 -(e") = e' 10 -(a') = a"lna 'dx 'dx d 1 d 1 II. - In Ix I= - 12. - (Iog" x) =- dx x dx x In a TRIGONOMETRIC FUNCTIONS d . d . . d ( ) , 13. - (SIn x) = cos x 14. -(cosx) = -SInX \/15. - tanx = sec-x dx dx dx d d d 16. dx (esc x) = -esc x cot x \./ 17. -(secx) = sec x tan x '/'18. - (cot x) = -csc ' x dx dx INVERSE -rRIGONOMETRIC FUNCTIONS d . 1 d 1 / d 1 19. dx (SIn-IX) = ~ \ / 20. - (cos-'x) = - ---=== V 21. - (tan-Ix) = - y 1 - x- / dr ~ dx 1 + x' d I d 1 d (' 1 22. - (esc-Ix) = -~ \/23. - sec- x) = ~ './ 24. - (coc'x) = ---, dx x x- - 1 . dx XyX- - I dx 1 + r HYPERBOLIC FUNCTIONS rd. d 25. ~ (sinh x) = cosh x 26. dx (cosh x) = smh x 1/ 27. - (tanh x) = sech'x''oj , dx d d d 28. dx (csch x) = -csch x coth x 29. - (sech x) = -sech x tanh x 30. - (coth x) = -csch'x dx dx INVERSE HYPERBOLIC FUNCTIONS d 1 d 1 d 1 31. - (sinh-Ix) = ~ 32. - (COSh-IX) = ,j 1 33. - (tanh-Ix) = - dx y 1 + x- dx x-I dx 1 - x' d -I 1 d 1 d 1 34. - (csch x) = - I I ~ 35. - (sech-Ix) = -~ 36. - (coth-'x) = --, dx x yX- + 1 dx x I-x dx 1 - x TABLE OF INTEGRALS BASIC FORMS S du I u 1. Su dv = uV - Svdu 6. Ssinudu = -cosu + C Jll. Scscucotudu = -cscu + C V16. =sin--+C ~ a S nd I n+1 C 2? U U =--u + . S du I I u n + I 7. Scosudu = sinu + C \Jl2. Stanudu = Inlsecul + C 'v17. -2--2 = - tan- - + C a+u a' a n>6 -I S du 3. ----;;=lnlul+C \/8. Ssec 2 udu = tanu + C J3. Scotudu = lnlsinul + C 4. Sen du = en + C V9. Scsc 2 udu = -cotu + C 1./14. Ssecudu = Inlsecu + tanu\ + C /5. Sandu = _I_a" + C lJO. Ssecutanudu = secu + C ViS. Scscudu = Inlcscu - cotul + C Ina FORMS INVOLVING .ja 2 + u 2 , a > 0 2 S u2du u a 21. Sla 2 + u 2 du = !!.. ~ + ~ In(u + Ja 2 + u' ) + C 26. = -~- - In(u + la 2 + u' ) + C 2 2 ~ 2 2 22. SU2~du = !!"(a' + 2u')~-~ In(u + .[(li +7) + C 8 8 27. S du = - 2. In I~+ a.1 + C u~ a u 23. ~ u du ~- a In la+~1 + C= u S du ~ 28 = - + C S .Ja'+u2 ~ . S a 2 U2~ u 24. 2 du=-+In(u+~)+C u .u S du . ~ 25. ~ = In(u + va2 + u') + C a' + u' FORMS INVOLVING .ja 2 - u 2 , a > 0 I 30. Sla 2 - u 2 du = !!..la 2 - u 2 +!.... sin-I!!.. + C 35. 'S du = --In la+~1 +c 2 2 a u~ a u S 4 2 U 2 a I u 31. u ~du = -(2u 2 - a )~ + ~ sin - - + C I 8 8 a --2-~+C au 32. S~ du=~-alnl a+~ I+C 2 3/2 U '2 2 ~ 3a 4 I u 37. S (a 2 -u) du=--(2u -5a)"a--u- +-sin--+C S ~ I u 8 8 a 33. -'--,:---du = --Ja 2 - u' - sin-'- + C u u a S u'du u a' I u 34. = --~ + - sin - - + C ~ 2 2 a FORMS INVOLVING .ju 2 - a 2 , a > 0 du 39. SJLi2Q-fdu = ~JLi2Q-f -~' lnlu + ~I + C 43. ---c:=~-= In Iu + Ju 2 - a 2 I + C S .J,;2=(if 40. Su2~du =i(2u 2 -a2)~ -~4 Inlu + ~I + C 44. S JU2 - a' . a 41. du = ~- acos-'- + C 45. u lui 2 2 S lu' - a Ju 2 - a 42. 2 du = - + In Iu + ..JUZ=aZ I + c 46. u u u --...:;...--+ C a2~ ....~111111111111111~-""---------------srf 'I 'I I udu I ,,2 du 2 , , , 47, ---= 2(a + bu - uln\a + bul) + C 56, ~ = --3(8a- + 3b-u - 4abu) ~+ C Ia + bu b - a + bu 15b I u'du I, 'I I 48, ---= -,[(a + bu)- - 4a(a + hu) + 2a In a + bu ] + C a + bu 2b 57. I ~ = ~ In I~ -~ I+ C, if a> 0 u a + bu va a + bu + a 49, r du ~ In I-_u-I + C 2 1{¥+bU= = --tan- --- + C if a < 0 ~ u(a + bu) a a + bu ..;=; -a' 50, I 'du I b Ia + bu I + c u\a + bu) = - -;;;; + a' In --u 58. I~ du = 2.JQ+!jU + a I ~ u u a + bu I udu a I 51. (a + bu)' = b'(a + bu) + I11n1u + bul + c 59. I ~ du = -~ +!!.- r du u' u 2 ~ u,;;;+b;i 52, I u(a ~u bu)' = a(a ~ bu) -~, In Ia : bu I + C 60. Iu"~ du = (2 [u"(a + bU)3 / ' - na rU"-I~ dU] b 2n + 3) ~ 53, I u'du, = -2, (a + bu - _a_,_ - 2a In Ia + bu I) + c (a + bu) b' a + bu u"du 2u"~ 211a I U,,-I du 54, I uJQ+bi7 du = ~(3bu - 2a)(a + bU)3/' + C 61. I .Ja + bu = b(2n + I) b(2n + I) ~ 15b du ~ b(211 - 3) I~ du 55. I udu '= ~(bu -2a)~ + C 62.. ~ r a(n - I)u" 1 2a(n - J) ~ U,,'I...fQ+b;; .Ja + bu 3b- - u" va + bu TRIGONOMETRIC FORMS 63, I sin'u du = ~ u -~ sin 2u + C 64, I cos'udu = ~u + ~ sin2u + C I ~ "d I ,,-, n - 2 I ,,-2 d ~ II - I n - I 77· sec u u = --tan u sec u + -- sec u u 65. I tan'u du =' tan u - u + C " -I ,,-, n - 2 I ,,-, 78. csc u du = --cot u csc u + -- csc u du I n - I n - I 66, I cot'udu = -cotu - u + C sin(a - b) u sin(a + b) u 79. sinausinhudu= ( ) - ( ) +C I 2a-b 2a+b 3 67. I sin u du = - t(2 + sin'u) cos u + C sin(a - b)u sin(a + b)u 80. cos au cos bu du = ( ) + ( ) + C I 2a-b 2a+b . 3 68. I cos udu = t(2 + cos,u)sinu + C cos(a-b)u cos(a + b) u 81. sin au cos bu du = - ----- ----+C I 2(a - b) 2(a + b) 3 69. I tan udu = ~ tan'u + In/cosul + C 82. I usinudu = sinu - ucosu + C 3 70. I cot u du = -~ cot'u - In Isin u I + C 83. I u cos u du = cos u + u sin u + C 3 71. I sec udu = ~ secutanu + t In/secu + lanul + C I ". d n I ,,-I d 84· U SIn u U = -u cos u + n u cos u u 3 72. I csc u du = -~ csc u cot u + t In Icsc u - cot u I + C d " ,,-I· dr I ."d 1. ,,-1 n - 1 I .,,-2 85 · I U " CDS U U = U Sin . u - n... U SIll U U 73 . SIn U U = - -;;- SIn u COS U + - n - SIn U du I J ... sin'l-lucosm+lu n - 1 r 1l~2 III I ,,-I· n I" - I ,,-, 86. sin"u cos"'u du = - + --- sin u cos u du 74. cos udu = -; cos USInU + -n- cos udu n+m n+m~ . 11+1 m-I sIn u cos u I I I~ I ." "'-' d75. tan"u du = --tan,,-J u - tan,,-2 u du m - I+ --- sm u cos u u n - I ~ n+m n + m f),t . -., ' ... ' ... ", .~".'<'*,"-, ¥Sf' ~" J' \ 1" ~ < .~. \.,( INVERSE TRIGONOMETRIC FORMS f 1 u 2 + I u 87. f sin-Iudu = usin-1u + ~ + C 92. u tan- udu = ---tan-'u - - + C 2 2 88. f cos-'udu = ucos-1u -~+ C 93. f u"sin-'udu = __l __ [u,,+lsin-l u - f U"+ldU._J, n,6-1 89. f f 90. f 91. tan-'udu = utan-1u -! In(l + u 2 ) + c 2u2-1 u~ usin-'udu = -- sin-1u + + C 4 4 2u2-1 u~ ucos-'udu=---cos-'u +c 4 4 94. 95. f f n+1 ~. ", I [,,+1 , f U"+I du J u cos - u du = - u cos - u + ~' n ,6 ,-1 n + I y 1- u 2 " -I I [n+1 - fUn+'dUJ u tan udu=- u tan 'u - --2' n,6-1 n+1 I+u EXPONENTIAL AND LOGARITHMIC FORMS 96. f uea" du = ~2 (au - l)e au + C t 100. flnudu=ulnu-u+C 97. f u"ea"du = ~u"ea" -~ f u"-'ea"du 101. f 98. f e""sinbudu = ~(asinbu - bcosbu) + C \ /102. f a + b v 99. f eO"cosbudu = ~(acosbu + bsinbu) + C a + b HYPERBOLIC FORMS \03. f sinh udu = cosh u + C !J04. f cosh u du = sinh u + C (": 105. f tanh u du = In cosh u + C 106. f cothudu = Inlsinhul + C [ 107. f sechudu = tan-'Isinhul + C FOR MS INVO LVING .,j2au - u 2 , a > 0 113. f .J2au - u 2 du = u; a .J2au _ u2 + ~ 114. u .J2au - u f 2 2au 115. f .J - u u f .J2au-u2 116. 2 u 117. f du 2 .J2au - u u du 118. _ f .j2a;:- u" 119. f, .. 'du 0 .J2au - u 2 du 120. -= = f U~U2 2u2-au-3a2 2 du = 6 .J2au . 108. f 109. f 110. f Ill. f 112. f cos-'( a: u) + C 3 a (a-u) - u 2 + 2: cos -I -a~ I du = .J2au _ u 2 + acos- ( a: u) + C 2.J2au-u2 _,(a-u) du = - - cos -- + C U a = COS-l(~)+ C a = -.j2au -- - u 2 + acos" (a -- - ") + C a 2 _!!! + ?:) .J2au _ u2 + 3a cos-1(a - u) + C 2 2 a 2 .J2au - u - + C au ,,+1 u"lnudu = u 2 [en + l)lnu - I] + C (n + I) _I- du '= In lIn uI + c ulnu cschudu = InJtanh!ul + C sech 2 u du = tanh u + C csch 2 u du = -coth u + C sech u tanh u du = -sech u +' C cschucothudu = -cschu + C