# Second Course in Calculus

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A SECOND COCi 1M YA IBS Rob  Constants tt  ~  3.141593   ir/2 « 1.570796   ir/3 « 1.047198   t/4 « 0.785398   x/6 « 0.523599   logio it « 0.497150   1 rad « 57.29578°1° » 0.0174533 rad   e « 2.718282   e2 » 7.389056   1/e « 0.367879   M = logio c « 0.4342945   1/Af = In 10 « 2.302585   logic M ~  0.637784 - 1   logio x = M  In x   V2  « 1.414214   V§ « 1.732051   V5 « 2.236068   30 mph = 44 ft/sec   g « 980.62 cm/sec2« 32.173 ft/sec2 Definite Integrals r2ir r2r  / sin2 nx dx =  / cos2 nx dx = t  (n > 1)    Jo Jo I 0  Jo ã2ir r2w sin mx sin nx dx —  / cos mx cos nx dx = 0    Jo(m n) sin mx cos nx dx = 03. f    sii    Jo r */2/ã? 4.  / sin2n xdx = Jo Jo ri r/2 /*ir/2 5. / sin2n+1 x dx =  / cos2n+1 x dx = Jo Jo   (2n + l)! 6. [ e ~x*dx = ^ Vtt    Jo 2  /2 ^ (2n)! *   cos2n x ax = ~■■ -   22n(n!)222 2n(n!)2 Formulas ax+1/= axav (a&)x = axbx(ax)v = axv   sin 21 + cos2 £= 1   1 + tan2£= sec2 £1 + cot22= csc2 £ 7T  7 r1— = cos - = — 63  2 7T   7 r  1 --  = cos - = — 442 7T  7 r1— = cos - = —362 1 /- sin( — 0 = — sin £cos( — 0 = cos t   sin(x + y)  = sin x cos y  +   cos(x + y)  = cos x cos y — sin 2x = 2 sin x cos x   cos 2x = cos2x — sin2 x   sin2 x = ^(1 — cos 2x)   cos2 x = i(l + cos 2x)   sinh t  = £(e* —e~l)   cosh t = + e“0   cosh2 £— sinh2 £= 1cos x sin 2  /    sin x sin 2  /  Power Series x| < 1 rb'X1' 1 n = 00 X xn— all x   n\n — 0 S X 2n+1  ‘ n *0 00 VA x2ncosz = 2/(-1)n(2^)! allx n =0 OO ln(l + *) = V (-1) -1— \x\ < 1 n n =1 arc tan x ■S' n =0(-1) X2n+1 2n -| 11*1< 1  Differentiation Rules m  I  I   cucuU  + Vu  + i)uvui)  + vuuvu — ui)VV2u[v(t)\ u[»(0]»(0 df . ,df   . = — x H --- y. dx dyy Differentiation Formulas m  £ - /«> ctael a1  In t sin £cos t  tan t  cot t  sec t  csc t arc sin t * arc tan t sinh t  cosh t  tanh t 0 at  “-1 el (In a) a1 1 t cos t  —   sin t   sec 21  —csc2 t   tan t   sec t  —cot t   CSC t 1Vi-11 + 1 2cosh t  sinh t  sech 21 Polar Coordinates { x —r  cos 0 y  = r  sin 0 Spherical CoordinatesIntegrals (constant of integration omitted)1.  J u dv  = uv — Jv du /v-M ; h 4 [ d±  = 1 J a2 — x2 2a  5. / dx  1 (x ——-= - arc tan I - + x2a \a,x  + a   x  — a In(a2+ x2)n  (2 n — 2 )a2(a2 + x2)n_1 2n  — 3 / dx<i-S [ ~  2) a2  J (n > 1)(2n - 2)a2  J  (a2+ x2) -1c f dx . (x\ 6. / —= arc sin 1- J J   Va2- x2W 7.  J y/a2— x2dx  = ^ \/a2— x2+ ~ arc sin8. J Vx2ia2y a/x2± a2 dx = In |x + \/x2± a29.= - Vx2± a2± In |x + \/x2± a2\ (continued inside back cover) Vectorsa ãb = dibi  -f- (I 2&2   I- C& 3&3 a x b =i jk a   1 &2 as 61 62 63 i  j-   dy + k dcte V / = grad / = /,)V ãv = div v = ux   + vy   + w. f x = p sin <t> cos 0 i jks y  = p sin <£sin 0 V X v = curl v = d/dx d/dyd/az i 2 = p COS 0 U  V  w

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