[PDF] Table of Basic Integrals Basic Forms



Previous PDF Next PDF







List of integrals of exponential functions

List of integrals of exponential functions 2 where where and is the Gamma Function when , , and when , , and Definite integrals for, which is the logarithmic mean



87 Table of Integrals

to integrate each of the following functions with respect to t a) et, b) e5t, c) t7, d) √ t, e) cos5t, f) e−t Answers 1 a) x2 2 + c, b) x7 7 + c, c) x−1 −1 + c = −x−1 + c, or −1 x + c, d) x−2 −2 + c = −1 2 x−2 + c, or − 1 2x2 +c, e) lnx+c, f) x3/2 3/2 +c = 2 3 x3/2 +c, g) x1/2 1/2 +c = 2x1/2 +c, h) 1 3 e3x +c, i) 1



Table of Integrals

e t 2 dt (51) xe xdx= (x 1)e (52) Z xeaxdx= x a 1 2 eax (53) Z x2exdx= x2 2x+ 2 ex (54) Z x2 eaxdx= x a 2x a2 + 2 a3 (55) Z 3exdx= 3 2 + 6 6 ex (56) Z xn eax d= x eax a n Z 1 (57) Z xneax dx= ( n1) an+1 [1 + n; ax]; where ( a;x) = Z 1 x ta 1e t dt (58) Z eax 2 dx= i p ˇ 2 p a erf ix p a (59) Z e ax 2 dx= p ˇ 2 p a erf x p a (60) Z xe ax 2 dx



Table of Integrals - University of Alberta

Table of Integrals Z sinaxsinbxdx = sin[(a−b)x] 2(a−b) − sin[(a+b)x] 2(a+b), (a26= b2) Z sin2axdx = x 2 − sin2ax 4a Z cosaxcosbxdx = sin[(a−b)x] 2(a−b





Table of Integrals

©2005 BE Shapiro Page 3 This document may not be reproduced, posted or published without permission The copyright holder makes no representation about the accuracy, correctness, or



Intégrales doubles [Correction]

exp(−(x2 + y2))dxdyetg(R) = ZZ B R exp(−(x2 + y2))dxdy a)Montrerqueg(R) 6 f(R) 6 g(R √ 2) b)Endéduirelavaleurde Z +∞ 0 e−t2dt Exercice 24 [ 02546 ] [Correction] SoitC(R) lequartdedisquex> 0,y> 0,x2 + y2 6 R2,R>0 a)Montrerque Z R 0 e−t2 dt 2 estcomprisentre ZZ C(R) e−x2−y2 dxdyet ZZ C(R √ 2) e−x2−y2 dxdy b)Calculer ZZ C



3 Contour integrals and Cauchy’s Theorem

curve given by r(t), a t b, then we can view r0(t) as a complex-valued curve, and then Z C f(z)dz= Z b a f(r(t)) r0(t)dt; where the indicated multiplication is multiplication of complex numbers (and not the dot product) Another notation which is frequently used is the following We denote a parametrized curve in the complex plane by z(t),



4 Improper Integrals - homeiitmacin

t R t f(x) dx= 0 for every t2R, but the integrals R c 1 f(x) dxand 1 c f(x) dxdo not exist for any c2R Next we consider integrals of functions de ned over in nite integrals of the form (a;1) and (1 ;b) De nition 4 2 (i) Suppose f is de ned on (a;1) and R 1 t f(x)dxexists for all t>a If lim ta Z 1 t f(x) dxexists, then we de ne the improper



Tempered distributions and the Fourier transform

18 1 TEMPERED DISTRIBUTIONS AND THE FOURIER TRANSFORM by (1 36) O K( )(˚) = Z K˚ dxdy: Theorem 1 2 There is a 1-1 correspondence between continuous linear oper-

[PDF] integrale sin(t)/t^2

[PDF] integrale sin(t)/t

[PDF] procédés théatraux

[PDF] tendinopathie genou traitement

[PDF] tendinite demi membraneux

[PDF] comment soigner une fabella

[PDF] fabella douloureuse

[PDF] tendinite poplité traitement

[PDF] mecanique de fluide resume

[PDF] mécanique des fluides bernoulli exercices corrigés

[PDF] fiche résumé mécanique des fluides

[PDF] formulaire mécanique des fluides pdf

[PDF] mécanique des fluides cours pdf

[PDF] question ? choix multiple culture générale

[PDF] question ? choix multiple definition

Table of Basic Integrals

Basic Forms

(1) Z x ndx=1n+ 1xn+1; n6=1 (2) Z1x dx= lnjxj (3) Z udv=uvZ vdu (4)

Z1ax+bdx=1a

lnjax+bj

Integrals of Rational Functions

(5)

Z1(x+a)2dx=1x+a

(6) Z (x+a)ndx=(x+a)n+1n+ 1;n6=1 (7) Z x(x+a)ndx=(x+a)n+1((n+ 1)xa)(n+ 1)(n+ 2) (8)

Z11 +x2dx= tan1x

(9) Z1a

2+x2dx=1a

tan1xa 1 (10) Zxa

2+x2dx=12

lnja2+x2j (11) Zx2a

2+x2dx=xatan1xa

(12) Zx3a

2+x2dx=12

x212 a2lnja2+x2j (13) Z1ax

2+bx+cdx=2p4acb2tan12ax+bp4acb2

(14)

Z1(x+a)(x+b)dx=1balna+xb+x; a6=b

(15)

Zx(x+a)2dx=aa+x+ lnja+xj

(16) Zxax

2+bx+cdx=12alnjax2+bx+cjba

p4acb2tan12ax+bp4acb2

Integrals with Roots

(17)

Zpxa dx=23

(xa)3=2 (18)

Z1pxadx= 2pxa

(19)

Z1paxdx=2pax

2 (20) Z xpxa dx=8 :2a3 (xa)3=2+25 (xa)5=2;or 23
x(xa)3=2415 (xa)5=2;or 215
(2a+ 3x)(xa)3=2 (21)

Zpax+b dx=2b3a+2x3

pax+b (22) Z (ax+b)3=2dx=25a(ax+b)5=2 (23)

Zxpxadx=23

(x2a)pxa (24) Z rx axdx=px(ax)atan1px(ax)xa (25) Z rx a+xdx=px(a+x)alnpx+px+a (26) Z xpax+b dx=215a2(2b2+abx+ 3a2x2)pax+b (27)

Zpx(ax+b)dx=14a3=2h

(2ax+b)pax(ax+b)b2lnapx+pa(ax+b)i (28) Zpx

3(ax+b)dx=b12ab28a2x+x3

px

3(ax+b)+b38a5=2lnapx+pa(ax+b)

(29) Zpx

2a2dx=12

xpx 2a212 a2lnx+px 2a2 3 (30) Zpa

2x2dx=12

xpa

2x2+12

a2tan1xpa 2x2 (31) Z xpx

2a2dx=13

x2a23=2 (32) Z1px

2a2dx= lnx+px

2a2 (33) Z1pa

2x2dx= sin1xa

(34) Zxpx

2a2dx=px

2a2 (35) Zxpa

2x2dx=pa

2x2 (36) Zx2px

2a2dx=12

xpx 2a212 a2lnx+px 2a2 (37) Zpax

2+bx+c dx=b+ 2ax4apax

2+bx+c+4acb28a3=2ln2ax+b+ 2pa(ax2+bx+c)

Z xpax

2+bx+c dx=148a5=2

2pa pax

2+bx+c3b2+ 2abx+ 8a(c+ax2)

+3(b34abc)lnb+ 2ax+ 2pa pax

2+bx+c(38)

4 (39) Z1pax

2+bx+cdx=1pa

ln2ax+b+ 2pa(ax2+bx+c) (40) Zxpax

2+bx+cdx=1a

pax

2+bx+cb2a3=2ln2ax+b+ 2pa(ax2+bx+c)

(41)

Zdx(a2+x2)3=2=xa

2pa 2+x2

Integrals with Logarithms

(42) Z lnax dx=xlnaxx (43) Z xlnx dx=12 x2lnxx24 (44) Z x

2lnx dx=13

x3lnxx39 (45) Z x nlnx dx=xn+1lnxn+ 11(n+ 1)2 ; n6=1 (46)

Zlnaxx

dx=12 (lnax)2 (47) Zlnxx

2dx=1x

lnxx 5 (48) Z ln(ax+b)dx= x+ba ln(ax+b)x;a6= 0 (49) Z ln(x2+a2)dx=xln(x2+a2) + 2atan1xa 2x (50) Z ln(x2a2)dx=xln(x2a2) +alnx+axa2x (51) Z lnax2+bx+cdx=1a p4acb2tan12ax+bp4acb22x+b2a+x lnax2+bx+c (52) Z xln(ax+b)dx=bx2a14 x2+12 x 2b2a 2 ln(ax+b) (53) Z xlna2b2x2dx=12 x2+12 x 2a2b 2 lna2b2x2 (54) Z (lnx)2dx= 2x2xlnx+x(lnx)2 (55) Z (lnx)3dx=6x+x(lnx)33x(lnx)2+ 6xlnx (56) Z x(lnx)2dx=x24 +12 x2(lnx)212 x2lnx (57) Z x

2(lnx)2dx=2x327

+13 x3(lnx)229 x3lnx 6

Integrals with Exponentials

(58) Z e axdx=1a eax (59) Zpxe axdx=1a pxe ax+ip

2a3=2erfipax

;where erf(x) =2p Z x 0 et2dt (60) Z xe xdx= (x1)ex (61) Z xe axdx=xa 1a 2 e ax (62) Z x

2exdx=x22x+ 2ex

(63) Z x

2eaxdx=x2a

2xa 2+2a 3 e ax (64) Z x

3exdx=x33x2+ 6x6ex

(65) Z x neaxdx=xneaxa na Z x n1eaxdx (66) Z x neaxdx=(1)na n+1[1 +n;ax];where (a;x) =Z 1 x ta1etdt (67) Z e ax2dx=ip 2 pa erfixpa 7 (68) Z e ax2dx=p 2 pa erfxpa (69) Z xe ax2dx=12aeax2 (70) Z x

2eax2dx=14

r a

3erf(xpa)x2aeax2

Integrals with Trigonometric Functions

(71) Z sinax dx=1a cosax (72) Z sin

2ax dx=x2

sin2ax4a (73) Z sin

3ax dx=3cosax4a+cos3ax12a

(74) Z sin nax dx=1a cosax2F112 ;1n2 ;32 ;cos2ax (75) Z cosax dx=1a sinax (76) Z cos

2ax dx=x2

+sin2ax4a (77) Z cos

3axdx=3sinax4a+sin3ax12a

8 (78) Z cos paxdx=1a(1 +p)cos1+pax2F11 +p2 ;12 ;3 +p2 ;cos2ax (79) Z cosxsinx dx=12 sin2x+c1=12 cos2x+c2=14 cos2x+c3 (80) Z cosaxsinbx dx=cos[(ab)x]2(ab)cos[(a+b)x]2(a+b);a6=b (81) Z sin

2axcosbx dx=sin[(2ab)x]4(2ab)+sinbx2bsin[(2a+b)x]4(2a+b)

(82)quotesdbs_dbs22.pdfusesText_28