MAT 113 Useful File Integrals Table, Fall 2005

A Brief Table of Integrals 1. J de = we ~ fedu 2. fera= "3, aFl, a>0 In a P 3 cost du = sinu + C 4. i sinidu = —cosu + C w . ut Chal - 8. [ (ax + by ae = SEM. ne-l 6. J (art by de =Finjax +b) +0 a(n + 1) a ff . (av + by" Tay tb b 7. | x(ax + by" dy = - - + #-1,-2 x(ax + by)" dy = m2 nel Cc n 1 a . 8. | x(ay + b)7 tv=8-Ainlavsb|+e 9. J sax tor Ady = Liters + b Jee @ a ax +b - [Gyr dv _ x . ’ = By" 2 (Var t by to 10. J = pin web 7f il. | overs by" dx = a nv? * C. ne 2 Nar ih — dx . 12, [Xt 2u sv art+s fe * xVax +b ly 2 -1 fax-b 13. (a) J £ = tan”! +C¢ xv ab Nb y 6 Vax+b-Vb w | ie = Ly i Na +C wVaxrtb Vb ING tb 45 is, { Marto 2 ath, ef dc 1s. f dx Nox tb a f de on; 7 Warts ‘SNertb br 26) Werth dx La if dx x 1k, 16. s =a ay . =—— Tt tan Cc J e+e a" a c u (24x 2a%(a? + x*) a a” dx 1 dx x 1 x 18. == .f == tt “inittelac ae 19 (a? =x?) 2aX(a?- x7) dah [X74 20. | tae side c= InQx + Vai tx) + C 21. [vera tVeTE Ener verte | " 4 22, fz Cre dv= ste + 2)Vai +x? - gin (et Var tx?) 4+C Vat 3 Vetx Vatx Vat + 23, [ YORE y= VEE ~ an [SEVOE| y c 2s. f VatE de =in(et Var te Metric ; ; ne : 2 ? A Ve 25. f Eade Lint Ve txry+ err ic Varre _2 26. [ dx = -j| jut Va? tx? +c 27. f dx = -V@ tee Werte « « “SD eves a’x “2 Ww uw A Brief Table of Integrals dk Va" = x r — = 4 J eVGA ax VE rade 5VE =a Sinjxt VE -a@|+c r _ ” Vaal dew na* Joe — ay"? dx, w n+] net .f dx Va = ae n-3 dx (Vit = a)" Qa? (n= 2a? 4 (Vx =a? > Veer J (Var = ary" av =! +C. n#-2 2 29, | Vat=Fac= 3 Vata 4 Sint E ac 32, f YEE ac = sin ¥- VEE yc x dx 1 atVa=-x a4. [ =—lin +C xVar =x? a x dx -1x lao 36. [ A = cosh 54+ C=infxt Vere Vinal a n#-t n#2 Vena k= Infxt Vea | - we 8S. | V2ax -— x? de = > {Wax + _ sin"! (: = 2) +C a V2 9 (Waar <2 y'de= (rm ay(V2ar = Ys na? | der = x7) de J n+] +7 +1 0. f dy - (x — a)(V2ax = x7" pe ied 3 dt J (V2ax - x*)" (a = 2)a? (n= 2)a? 4S (Vax — 8)? r. xt a)Qv-3a)Via— xg’ - 1. ] xV2ax — xi dr = Ba SONNE + sin7! (3 as Cc ( V2ax— x2 - Var — - 2 | M2 TE te = Van = Ft asin (E52) +¢ 53. J soe x 2] 2a - x 4. fae ng gig! 324) _ Vaax- 4c 55. [Ss 2a =k 4 J Via = x? a —— ae 6. J sinavas = ~heosart C 57. [ fuea ee 8 as aX _ sin2av « J x, sin Zax . / sin’ ax dx 5 da +C 59, | costardr = 3 qa +C l+¢ 7 : , sin’! ' _ —_ nl oy cin ar - 60. J sin" ax dx = — Sin _ ae 60S OR + nat f sin’? ax dx 61. | cos" ax dx = sos arsine + aot} cos"? ax dx : cos(at+b)x cos(a~ b)x 62. (af sinaxcosbedx = SOS — Sah 2 tb a +) Tansy 7 8 P : + sin(a— b)x — sin(a + b)x (b) J sin axsin beds = Se = Soaht Pzb? Sanh ath 7° b sin(a—b)x , sin(a + b)x (c) J c0s ax cos bx a = tH ead OX = Fa = b) Yatb) web . Jax «ntl oy 63. J sin ax cos ax dx = SSAC 64. J sin” ax cos ax dx = sin 8 +, n#—-1 4a (a + Ia " nel 6s, { Eda} oo J none _ cos"! ax ae Snax t 7a In| sinax]+C 66. | cos"axsinax dx (re ia +O, n#-l in 67. { sn de = -1 In| cosax|+C sanml m+) ny _ . 68. J sin” ax cos” ax dx = Sin ax cos ak nol sin"? ax cos” ax dx, neo-m (reduces sin" ax) am + 1) mtn . sat lav cos"! ay - 5 69. | sin™ ax cos” ax dx = sin’ 6060s Ox 5 mao sin" ax cos”? ax dx, mez (reduces cos” ax) a(n + 1) mon ff dx -2 b- wax 70. . = =— tan™! fin (Z-#)l 4. Boe Jbtesinax avec bee "\4 2 , . i. f dx ert yy fe tb sin ax Ver = b? cos ax} ic Bee .J btesinay aver b+c sin ax : dx 1 wT _ ax J dx az. ax 72. ; = -stanf2-S)+ | —a— egestas] e J 1+ sin ax Lian(3 2 c 3 1 — sin ax fant 2 c 14. [ dx = 2 tan”! b= 6g Ble b>e Jb+cecosax eVvbt-c bre 2 75. f ax = 1 in | ob £08 ax + Ver = b* sin atl ac Bed Jbrccosax aVvei-b b +c cos ax : dx 1 ax dx ax 1s. [ cefie = hen eo Lag > 1+ cos ax gant 7. | 1 — cos ax acts c P P . 78. | xsinaxdx = 4 sin ax — Scosax + C 79. | xcosaxdx = ae cos ax + Ssinavt+ C J a a J ay a ‘a “LU ft a” 80. | x"sinax dx = = cos ax + a J x" cos ax dx 81. | x"cosaxdx = eS sin ax ~ af x"! sin ax dx J P 82. J tan ax dv = din |secax|+C §3. | cotaxdy= 4 in| sin ax| +C c c 84. | tan? ax de = tan ax ~x+C 85. | cotarde= —deotar—x+C J ‘a and -1 86. J tan" ax dx = nT) - J tan’? ax dx, n=l 87. J cot" ax dx = ay - J cot’? ax dx, n#1 r 8S. | secardx = din |secax + tanax|+C 89. J esc ax dx = lin |ese ax +cotax]+C J P 1 1 90. | sec? ax dx = Ztanart+C 91. J esc? axde= ~geotar + -2 ec” “ax tan ax - 2 5 | sec" ax dx = sec” “ax tan ax 4 2 ; { sec"™* ax dx, n- a(n — 1) ~ A Brief Table of Integrals T-3 ‘T-4 A Brief Table of Integrals a=2 93. J cscraxas = —SSC__ ak Cot ae) nm 2 esc" ax dr, n#1 a(n — 1) n-1 ft a 94. J sec" ax tanax dx = Sec’ ax +C, n#0 95. J esc" ax cotaxdr = See +C, n#0 r 96. ] sin”! axdx = x sin“ ax + dvi =a tc 97. J cos"! ax de = x cos" ax — lV -@e+tC r 98. J tan”! ax dy = x tan“! ar — xa +aC)4C ft nel a 99. nett = 2 on xt de ee jor" sin axdx nary net) Vrogh? n 1 f na n=l 100. | x"cos"ardr == Vay +— J xd #- J axdx pare ao ax Vict n 1 net nl 101. f "tanlaxde = 4 “gy 4 xd - x" tan” ax dx hei ax n+l) Woreht n#-1 ‘ c ax 102, | dra bere 103. f or de= 2 bY 4 ¢, b>0, b#1 fos J setae = Sea - N+tc \ 105. gute - f 7 apa 106. | x"b“ dr = = -— aml pae = . 6. | x =o anos? bY dx, b>0. b#1 107 108. J e* cos bx dx = — B (a cos bx + b sin bx) + C 109. il Inax dx =xInax —x+C ath: aa, mn In ax 110. J x"(In ax)" dx = oe - TI x“Unaxy"! de. n#-t 1 xy! 1. fx tdnanr ae = 22Y™ 5. ¢, m#-1 2. f ae = in|inax|+C mtt xin ax 113, J sinh ax de = + cosh ax + C 114, { cosh av dx = 4 sinh av + C 15, J sinh? ax de = HOh2ax_ <4 116. if cosh? ax dy = Sih2ar 4 X 4 ¢ 4a 2 da 2 bel ee . — 117. if sinh” ax dx = sinh’ ax cosh ax -4 1 l sinh"~? av dv. n#0 S18. | cosh" ax dx = Sosh ax sinh ax +4 = u | cosh”™* av dy, n#0 119. Js sinh ax dx = 2 cosh ax — ae sinhax + C 120. J x cosh ax dy = * sinh ax - = cosh ant a a 121. fe sinh ax dx = x cosh ax ~ nf x"! cosh ax dx 122. | x" cosh ax dv = a sinh ax — af x"! sinh ax de 123. if tanh ax dx = 1 In (cosh ax) + C 124. J coth ax de = 4 In| sinhax| +C 125, J tanh? ax dr = x — 1 tanhax+C 126. J coth? ax dx =x - doth ax+C ant 127, J tanh" ax dx = ea + i tanh"? ax dr, n#l a-l 128, il coth" ax dx = ee + J coth"axdy, n#1 129, J sech ardx = J gin! (tanh ax) + C 130. J esch ax dx = din tanh F +C