Table of Integrals

Basic Forms

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

1ax+bdx=1a

lnjax+bj(4)

Integrals of Rational Functions

1(x+a)2dx=1x+a(5)

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

11 +x2dx= tan1x(8)

Z 1a

2+x2dx=1a

tan1xa (9) Z xa

2+x2dx=12

lnja2+x2j(10) Z x2a

2+x2dx=xatan1xa

(11) Z x3a

2+x2dx=12

x212 a2lnja2+x2j(12) Z 1ax

2+bx+cdx=2p4acb2tan12ax+bp4acb2(13)

1(x+a)(x+b)dx=1balna+xb+x; a6=b(14)

Z x(x+a)2dx=aa+x+ lnja+xj(15) Z xax

2+bx+cdx=12alnjax2+bx+cj

ba p4acb2tan12ax+bp4acb2(16)

Integrals with Roots

Z pxadx=23 (xa)3=2(17) Z

1pxadx= 2pxa(18)

1paxdx=2pax(19)

Z xpxadx=23 a(xa)3=2+25 (xa)5=2(20) Z pax+bdx=2b3a+2x3 pax+b(21) Z (ax+b)3=2dx=25a(ax+b)5=2(22) Z xpxadx=23 (x2a)pxa(23) Z rx axdx=px(ax)atan1px(ax)xa(24) Z rx a+xdx=px(a+x)alnpx+px+a(25)Z xpax+bdx=215a2(2b2+abx+ 3a2x2)pax+b(26) Z px(ax+b)dx=14a3=2h (2ax+b)pax(ax+b) b2lnapx+pa(ax+b)i (27) Z px

3(ax+b)dx=b12ab28a2x+x3

3(ax+b)

b38a5=2lnapx+pa(ax+b)(28) Z px

2a2dx=12

xpx 2a212 a2lnx+px 2a2 (29) Z pa

2x2dx=12

xpa

2x2+12

a2tan1xpa 2x2 (30) Z xpx

2a2dx=13

x2a23=2(31) Z 1px

2a2dx= lnx+px

2a2(32)

Z 1pa

2x2dx= sin1xa

(33) Z xpx

2a2dx=px

2a2(34)

Z xpa

2x2dx=pa

2x2(35)

Z x2px

2a2dx=12

xpx 2a212 a2lnx+px 2a2 (36) Z pax

2+bx+cdx=b+ 2ax4apax

2+bx+c

4acb28a3=2ln2ax+b+ 2pa(ax2+bx+c)(37)

Z xpax

2+bx+c=148a5=2

2pa pax

2+bx+c

3b2+ 2abx+ 8a(c+ax2)

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

2+bx+c

(38) Z 1pax

2+bx+cdx=1pa

ln2ax+b+ 2pa(ax2+bx+c) (39) Z xpax

2+bx+cdx=1a

pax

2+bx+c

b2a3=2ln2ax+b+ 2pa(ax2+bx+c)(40) Z dx(a2+x2)3=2=xa 2pa

2+x2(41)Integrals with Logarithms

Z lnaxdx=xlnaxx(42) Z lnaxx dx=12 (lnax)2(43) Z ln(ax+b)dx= x+ba ln(ax+b)x;a6= 0 (44) Z ln(x2+a2) dx =xln(x2+a2) + 2atan1xa

2x(45)

Z ln(x2a2) dx =xln(x2a2) +alnx+axa2x(46) Z lnax2+bx+cdx=1a p4acb2tan12ax+bp4acb2

2x+b2a+x

lnax2+bx+c(47) Z xln(ax+b)dx=bx2a14 x2 12 x 2b2a 2 ln(ax+b) (48) Z xlna2b2x2dx=12 x2+ 12 x 2a2b 2 lna2b2x2(49)

Integrals with Exponentials

Z e axdx=1a eax(50) Z pxe axdx=1a pxe ax+ip

2a3=2erfipax

where erf(x) =2p Z x 0 et2dt(51) Z xe xdx= (x1)ex(52) Z xe axdx=xa 1a 2 e ax(53) Z x

2exdx=x22x+ 2ex(54)

Z x

2eaxdx=x2a

2xa 2+2a 3 e ax(55) Z x

3exdx=x33x2+ 6x6ex(56)

Z x neaxdx=xneaxa na Z x n1eaxdx(57) Z x neaxdx=(1)na n+1[1 +n;ax]; where (a;x) =Z 1 x ta1etdt(58) Z e ax2dx=ip 2 pa erfixpa (59) Z e ax2dx=p 2 pa erfxpa (60) Z xe ax2dx =12aeax2(61) Z x

2eax2dx =14

r a

3erf(xpa)x2aeax2(62)?

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Integrals with Trigonometric Functions

Z sinaxdx=1a cosax(63) Z sin

2axdx=x2

sin2ax4a(64) Z sin naxdx= 1a cosax2F112 ;1n2 ;32 ;cos2ax (65) Z sin

3axdx=3cosax4a+cos3ax12a(66)

Z cosaxdx=1a sinax(67) Z cos

2axdx=x2

+sin2ax4a(68) Z cos paxdx=1a(1 +p)cos1+pax 2

F11 +p2

;12 ;3 +p2 ;cos2ax (69) Z cos

3axdx=3sinax4a+sin3ax12a(70)

Z (71) Z sin

2axcosbxdx=sin[(2ab)x]4(2ab)

sinbx2bsin[(2a+b)x]4(2a+b)(72) Z sin

2xcosxdx=13

sin3x(73) Z cos

2axsinbxdx=cos[(2ab)x]4(2ab)cosbx2b

cos[(2a+b)x]4(2a+b)(74) Z cos

2axsinaxdx=13acos3ax(75)

Z sin

2axcos2bxdx=x4

sin2ax8asin[2(ab)x]16(ab) sin2bx8bsin[2(a+b)x]16(a+b)(76) Z sin

2axcos2axdx=x8

sin4ax32a(77) Z tanaxdx=1a lncosax(78) Z tan

2axdx=x+1a

tanax(79) Z tanquotesdbs_dbs6.pdfusesText_12

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