TablesofIntegralsME501F2011

TablesofIntegralsME501F2011 - 411 CALCULUS INTEGRALS...

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411 CALCULUS INTEGRALS (Contiuued) dx J ax 2 + c (m - 2)a f dx 284. - - --- f - 1 xmFx2 + c - 1)cx m - 1)c x m - Z Jax 2 + c 1 + x 2 1 xfi + Ji+ X4 285. f = -log ----'---~,___- (1 - x 2 )Jl + X4 f X Z 1 - 286. _ = (1 + x 2 )J1+7 f 2 287. = - xJx" + a 2 na f 2 . -1 a 288 ---- = - -SIn -- . xft - a 2 ft 2 (x)t 289. fj - 3 x 3 = - sin - 1 - a - x 3 a FORMS INVOLVING TRIGONOMETRIC FUNCTIONS 290. f (sin ax) dx 291. f (cos I 292. (tan 293. f (eot 294. f (sec 295. (esc ax) dx I 296. I (sin2 = 29.7. (sin 3 = d f 298 . f (sm . 4 ax) x = 299. I (sm" . = x 2 fi 1 - 1 - tan - 1 -;======;c Jl + a + Jx" + a 2 = - ~ cos ax l. = ~ sm II = - - log cos = - log sec a a = ~ log sin = - ~ log esc a a = ~log(seeax + tan ax) =~IOgtan(~ + a a 4 2 ~ = - log (esc - cot = tan- a a 2 - 1 - cos sin + -x = - 1 - 2ax 2a 2 2 4a - -(cos ax)(sm 2 + 2) 3a 3x - - sin sin 4ax + 8 32a sin" - 1 n - 1 f - + (sin"-2 n
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412 300 301. 302. 303. 304. 305. 3 06 . 307. 308. 309. 310. 311. 312. 313. 314. d SIn m ax x = - f ( . 2 ) f . (sm2m+ 1 ax) dx = f (cos 2 ax)dx f 3 f 4 d (cos ax) x f CALCULUS INTEGRALS (Continued) COS m I -1 (2m)!(r!)2 . 2 + 1 (2m)! -- sm' x a ,=0 2 2m - 2 '(2r + 1)!(m!)2 + 22m(m!f COS m 2 - 2 '(m!)2(2r)! - I sin 2 ' a (2m + l)!(r!f (cos"ax)dx =-cos"-laxsinax+-- - (cos"-2ax)dx na n 2 d sin m - 1 (2m) !(r !)2 2 + 1 (2m)! f cos m x = I cos' x ( ) a 2 - 2 + 1)!(m!)2 + 22m(m!)2 m 22m-2r(m!)2(2r)! f + 1 = I cos 2r a r=O + l. 1 1 1 = -sin cos + -x = + -sin2ax 2a 2 2 4a = -(sm ax)(cos 2 + 2) 3a 3x 2ax 4ax = - + + -2- 8 3 a In-If f dx f 1 . 2 = (csc 2 = - - cot ~n a f f 1 = (csc m = . m (m - l)a 1 f f = (csc = - - cos f a - f 2m+l - . 2 +1 - (esc ax)dx - sm m m - 2 f + m 1 m - 1 m - 2 m-122m-2r-l(m - 1)!m!(2r)! r~o (2m)!(r!)2sin2r+1ax 1 m-l (2m)!(r!)2 1 (2m)! - - cos ~ + - . log tan - a ,~o 2 - 2 '(m!)2(2r + I)! sin2r+2 a 22m(m!)2 2 f f 1 2 = (sec 2 = - tan a f 1 n - 2 f dx f = (sec" = . + cos" (n - cos"- 1 n - 1 cos"- 2 _ 1 m-1 2m-2,-1 - f (sec = - I 2 - a - - - f
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CALCULUS 413 INTEGRALS (Continued) 315. I dx = I(sec 2m + 1 ax) dx = cos + 1 ax 1. m- 1 (2m) !(r !)2 - smax L a r=O 22m-2r(m!)2(2r + I)! COS 2 r+ 2 1 (2m)! + -. 2 2 log (sec + tan ax) a 2 m(m!) . sin(m - n)x sin(m + n)x 316. I (sin mxHsm nx) dx = - 2 ' 2(m - n) (m + n) + 317. I (cos mx)(cos = + 2 ' 2(m - + 318. I (sin ax)(cos ax) dx = ~ sin 2 2a cos(m - cos(m + 319. I (sin mx)(cos = 2(m - 2(m + 320. I(Sin 2 ax)(cos 2 = - _1_ sin 4ax + ~ 32a 8 cos m + 1 321. I (sinax)(cos m = + l)a sinm+ 1 322. I (sin'" = ---- + cos m - 1 sin" + 1 + n)a m -II + -- (cosm- 2 ax)(sin" ax) dx m + n 323. I (cos'" = or sin"-l ax cos m + 1 + n)a n- 1 I + (COS m ax)(sin"- 2 m+n cos'" + 1 m - n + 2 I cos'" - - (n - sin"- 1 n - 1 sin" 2 324. ICos'" ax dx = sin" COS",-l m - 1 Icos m - 2 -----~- + a(m - sin" 1 m - n sin"
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414 CALCULUS
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TablesofIntegralsME501F2011 - 411 CALCULUS INTEGRALS...

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