Table of Basic Integrals
Basic Forms
∫xndx=1n+1xn+1,n≠−1
∫xndx=1n+1xn+1,n≠−1
(1)
∫1xdx=ln∣∣∣x∣∣∣
∫1xdx=ln|x|
(2)
∫udv=uv−∫vdu
∫udv=uv−∫vdu
(3)
∫1ax+bdx=1aln∣∣∣ax+b∣∣∣
∫1ax+bdx=1aln|ax+b|
(4)
Integrals of Rational Functions
∫1(x+a)2dx=−1x+a
∫1(x+a)2dx=−1x+a
(5)
∫(x+a)ndx=(x+a)n+1n+1,n≠−1
∫(x+a)ndx=(x+a)n+1n+1,n≠−1
(6)
∫x(x+a)ndx=(x+a)n+1((n+1)x−a)(n+1)(n+2)
∫x(x+a)ndx=(x+a)n+1((n+1)x−a)(n+1)(n+2)
(7)
∫11+x2dx=tan−1x
∫11+x2dx=tan−1x
(8)
∫1a2+x2dx=1atan−1xa
∫1a2+x2dx=1atan−1xa
(9)
∫xa2+x2dx=12ln∣∣∣a2+x2∣∣∣
∫xa2+x2dx=12ln|a2+x2|
(10)
∫x2a2+x2dx=x−atan−1xa
∫x2a2+x2dx=x−atan−1xa
(11)
∫x3a2+x2dx=12x2−12a2ln∣∣∣a2+x2∣∣∣
∫x3a2+x2dx=12x2−12a2ln|a2+x2|
(12)
∫1ax2+bx+cdx=24ac−b2−−−−−−−√tan−12ax+b4ac−b2−−−−−−−√
∫1ax2+bx+cdx=24ac−b2tan−12ax+b4ac−b2
(13)
∫1(x+a)(x+b)dx=1b−alna+xb+x, a≠b
∫1(x+a)(x+b)dx=1b−alna+xb+x, a≠b
(14)
∫x(x+a)2dx=aa+x+ln∣∣∣a+x∣∣∣
∫x(x+a)2dx=aa+x+ln|a+x|
(15)
∫xax2+bx+cdx=12aln∣∣∣ax2+bx+c∣∣∣−ba4ac−b2−−−−−−−√tan−12ax+b4ac−b2−−−−−−−√
∫xax2+bx+cdx=12aln|ax2+bx+c|−ba4ac−b2tan−12ax+b4ac−b2
(16)
Integrals with Roots
∫x−a−−−−−√dx=23(x−a)3∕2
∫x−adx=23(x−a)3∕2
(17)
∫1x±a−−−−−√dx=2x±a−−−−−√
∫1x±adx=2x±a
(18)
∫1a−x−−−−−√dx=−2a−x−−−−−√
∫1a−xdx=−2a−x
(19)
∫xx−a−−−−−√dx=⎧⎩⎨⎪⎪⎪⎪⎪⎪2a3(x−a)3∕2+25(x−a)5∕2, or23x(x−a)3∕2−415(x−a)5∕2, or215(2a+3x)(x−a)3∕2
∫xx−adx=2a3x−a3∕2+25x−a5∕2, or23x(x−a)3∕2−415(x−a)5∕2, or215(2a+3x)(x−a)3∕2
(20)
∫ax+b−−−−−√dx=(2b3a+2x3)ax+b−−−−−√
∫ax+bdx=2b3a+2x3ax+b
(21)
∫(ax+b)3∕2dx=25a(ax+b)5∕2
∫(ax+b)3∕2dx=25a(ax+b)5∕2
(22)
∫xx±a−−−−−√dx=23(x∓2a)x±a−−−−−√
∫xx±adx=23(x∓2a)x±a
(23)
∫xa−x−−−−−√dx=−x(a−x)−−−−−−−√−atan−1x(a−x)−−−−−−−√x−a
∫xa−xdx=−x(a−x)−atan−1x(a−x)x−a
(24)
∫xa+x−−−−−√dx=x(a+x)−−−−−−−√−aln[x−−√+x+a−−−−−√]
∫xa+xdx=x(a+x)−alnx+x+a
(25)
∫xax+b−−−−−√dx=215a2(−2b2+abx+3a2x2)ax+b−−−−−√
∫xax+bdx=215a2(−2b2+abx+3a2x2)ax+b
(26)
∫x(ax+b)−−−−−−−−√dx=14a3∕2[(2ax+b)ax(ax+b)−−−−−−−−−√−b2ln∣∣ax−−√+a(ax+b)−−−−−−−−√∣∣]
∫x(ax+b)dx=14a3∕2(2ax+b)ax(ax+b)−b2lnax+a(ax+b)
(27)
∫x3(ax+b)−−−−−−−−−√dx=[b12a−b28a2x+x3]x3(ax+b)−−−−−−−−−√+b38a5∕2ln∣∣ax−−√+a(ax+b)−−−−−−−−√∣∣
∫x3(ax+b)dx=b12a−b28a2x+x3x3(ax+b)+b38a5∕2lnax+a(ax+b)
(28)
∫x2±a2−−−−−−√dx=12xx2±a2−−−−−−√±12a2ln∣∣x+x2±a2−−−−−−√∣∣
∫x2±a2dx=12xx2±a2±12a2lnx+x2±a2
(29)
∫a2−x2−−−−−−√dx=12xa2−x2−−−−−−√+12a2tan−1xa2−x2−−−−−−√
∫a2−x2dx=12xa2−x2+12a2tan−1xa2−x2
(30)
∫xx2±a2−−−−−−√dx=13(x2±a2)3∕2
∫xx2±a2dx=13x2±a23∕2
(31)
∫1x2±a2−−−−−−√dx=ln∣∣x+x2±a2−−−−−−√∣∣
∫1x2±a2dx=lnx+x2±a2
(32)
∫1a2−x2−−−−−−√dx=sin−1xa
∫1a2−x2dx=sin−1xa
(33)
∫xx2±a2−−−−−−√dx=x2±a2−−−−−−√
∫xx2±a2dx=x2±a2
(34)
∫xa2−x2−−−−−−√dx=−a2−x2−−−−−−√
∫xa2−x2dx=−a2−x2
(35)
∫x2x2±a2−−−−−−√dx=12xx2±a2−−−−−−√∓12a2ln∣∣x+x2±a2−−−−−−√∣∣
∫x2x2±a2dx=12xx2±a2∓12a2lnx+x2±a2
(36)
∫ax2+bx+c−−−−−−−−−−√dx=b+2ax4aax2+bx+c−−−−−−−−−−√+4ac−b28a3∕2ln∣∣2ax+b+2a(ax2+bx+c)−−−−−−−−−−−−√∣∣
∫ax2+bx+cdx=b+2ax4aax2+bx+c+4ac−b28a3∕2ln2ax+b+2a(ax2+bx+c)
(37)
∫xax2+bx+c−−−−−−−−−−√dx=148a5∕2(2a−−√ax2+bx+c−−−−−−−−−−√(−3b2+2abx+8a(c+ax2))+3(b3−4abc)ln∣∣b+2ax+2a−−√ax2+bx+c−−−−−−−−−−√∣∣)
∫xax2+bx+cdx=148a5∕22aax2+bx+c−3b2+2abx+8a(c+ax2)+3(b3−4abc)lnb+2ax+2aax2+bx+c
(38)
∫1ax2+bx+c−−−−−−−−−−√dx=1a−−√ln∣∣2ax+b+2a(ax2+bx+c)−−−−−−−−−−−−−√∣∣
∫1ax2+bx+cdx=1aln2ax+b+2a(ax2+bx+c)
(39)
∫xax2+bx+c−−−−−−−−−−√dx=1aax2+bx+c−−−−−−−−−−√−b2a3∕2ln∣∣2ax+b+2a(ax2+bx+c)−−−−−−−−−−−−−√∣∣
∫xax2+bx+cdx=1aax2+bx+c−b2a3∕2ln2ax+b+2a(ax2+bx+c)
(40)
∫dx(a2+x2)3∕2=xa2a2+x2−−−−−−√
∫dx(a2+x2)3∕2=xa2a2+x2
(41)
Integrals with Logarithms
∫lnaxdx=xlnax−x
∫lnaxdx=xlnax−x
(42)
∫xlnxdx=12x2lnx−x24
∫xlnxdx=12x2lnx−x24
(43)
∫x2lnxdx=13x3lnx−x39
∫x2lnxdx=13x3lnx−x39
(44)
∫xnlnxdx=xn+1(lnxn+1−1(n+1)2),n≠−1
∫xnlnxdx=xn+1lnxn+1−1(n+1)2,n≠−1
(45)
∫lnaxxdx=12(lnax)2
∫lnaxxdx=12lnax2
(46)
∫lnxx2dx=−1x−lnxx
∫lnxx2dx=−1x−lnxx
(47)
∫ln(ax+b)dx=(x+ba)ln(ax+b)−x,a≠0
∫ln(ax+b)dx=x+baln(ax+b)−x,a≠0
(48)
∫ln(x2+a2)dx=xln(x2+a2)+2atan−1xa−2x
∫ln(x2+a2)dx=xln(x2+a2)+2atan−1xa−2x
(49)
∫ln(x2−a2)dx=xln(x2−a2)+alnx+ax−a−2x
∫ln(x2−a2)dx=xln(x2−a2)+alnx+ax−a−2x
(50)
∫ln(ax2+bx+c)dx=1a4ac−b2−−−−−−−√tan−12ax+b4ac−b2−−−−−−−√−2x+(b2a+x)ln(ax2+bx+c)
∫lnax2+bx+cdx=1a4ac−b2tan−12ax+b4ac−b2−2x+b2a+xlnax2+bx+c
(51)
∫xln(ax+b)dx=bx2a−14x2+12(x2−b2a2)ln(ax+b)
∫xln(ax+b)dx=bx2a−14x2+12x2−b2a2ln(ax+b)
(52)
∫xln(a2−b2x2)dx=−12x2+12(x2−a2b2)ln(a2−b2x2)
∫xlna2−b2x2dx=−12x2+12x2−a2b2lna2−b2x2
(53)
∫(lnx)2dx=2x−2xlnx+x(lnx)2
∫(lnx)2dx=2x−2xlnx+x(lnx)2
(54)
∫(lnx)3dx=−6x+x(lnx)3−3x(lnx)2+6xlnx
∫(lnx)3dx=−6x+x(lnx)3−3x(lnx)2+6xlnx
(55)
∫x(lnx)2dx=x24+12x2(lnx)2−12x2lnx
∫x(lnx)2dx=x24+12x2(lnx)2−12x2lnx
(56)
∫x2(lnx)2dx=2x327+13x3(lnx)2−29x3lnx
∫x2(lnx)2dx=2x327+13x3(lnx)2−29x3lnx
(57)
Integrals with Exponentials
∫eaxdx=1aeax
∫eaxdx=1aeax
(58)
∫x−−√eaxdx=1ax−−√eax+iπ−−√2a3∕2erf(iax−−√), where erf(x)=2π−−√∫x0e−t2dt
∫xeaxdx=1axeax+iπ2a3∕2erfiax, where erf(x)=2π∫0xe−t2dt
(59)
∫xexdx=(x−1)ex
∫xexdx=(x−1)ex
(60)
∫xeaxdx=(xa−1a2)eax
∫xeaxdx=xa−1a2eax
(61)
∫x2exdx=(x2−2x+2)ex
∫x2exdx=x2−2x+2ex
(62)
∫x2eaxdx=(x2a−2xa2+2a3)eax
∫x2eaxdx=x2a−2xa2+2a3eax
(63)
∫x3exdx=(x3−3x2+6x−6)ex
∫x3exdx=x3−3x2+6x−6ex
(64)
∫xneaxdx=xneaxa−na∫xn−1eaxdx
∫xneaxdx=xneaxa−na∫xn−1eaxdx
(65)
∫xneaxdx=(−1)nan+1Γ[1+n,−ax], where Γ(a,x)=∫∞xta−1e−tdt
∫xneaxdx=(−1)nan+1Γ[1+n,−ax], where Γ(a,x)=∫x∞ta−1e−tdt
(66)
∫eax2dx=−iπ−−√2a−−√erf(ixa−−√)
∫eax2dx=−iπ2aerfixa
(67)
∫e−ax2dx=π−−√2a−−√erf(xa−−√)
∫e−ax2dx=π2aerfxa
(68)
∫xe−ax2dx=−12ae−ax2
∫xe−ax2dx=−12ae−ax2
(69)
∫x2e−ax2dx=14πa3−−−√erf(xa−−√)−x2ae−ax2
∫x2e−ax2dx=14πa3erf(xa)−x2ae−ax2
(70)
Integrals with Trigonometric Functions
∫sinaxdx=−1acosax
∫sinaxdx=−1acosax
(71)
∫sin2axdx=x2−sin2ax4a
∫sin2axdx=x2−sin2ax4a
(72)
∫sin3axdx=−3cosax4a+cos3ax12a
∫sin3axdx=−3cosax4a+cos3ax12a
(73)
∫sinnaxdx=−1acosax2F1[12,1−n2,32,cos2ax]
∫sinnaxdx=−1acosax2F112,1−n2,32,cos2ax
(74)
∫cosaxdx=1asinax
∫cosaxdx=1asinax
(75)
∫cos2axdx=x2+sin2ax4a
∫cos2axdx=x2+sin2ax4a
(76)
∫cos3axdx=3sinax4a+sin3ax12a
∫cos3axdx=3sinax4a+sin3ax12a
(77)
∫cospaxdx=−1a(1+p)cos1+pax×2F1[1+p2,12,3+p2,cos2ax]
∫cospaxdx=−1a(1+p)cos1+pax×2F11+p2,12,3+p2,cos2ax
(78)
∫cosxsinxdx=12sin2x+c1=−12cos2x+c2=−14cos2x+c3
∫cosxsinxdx=12sin2x+c1=−12cos2x+c2=−14cos2x+c3
(79)
∫cosaxsinbxdx=cos[(a−b)x]2(a−b)−cos[(a+b)x]2(a+b),a≠b
∫cosaxsinbxdx=cos[(a−b)x]2(a−b)−cos[(a+b)x]2(a+b),a≠b
(80)
∫sin2axcosbxdx=−sin[(2a−b)x]4(2a−b)+sinbx2b−sin[(2a+b)x]4(2a+b)
∫sin2axcosbxdx=−sin[(2a−b)x]4(2a−b)+sinbx2b−sin[(2a+b)x]4(2a+b)
(81)
∫sin2xcosxdx=13sin3x
∫sin2xcosxdx=13sin3x
(82)
∫cos2axsinbxdx=cos[(2a−b)x]4(2a−b)−cosbx2b−cos[(2a+b)x]4(2a+b)
∫cos2axsinbxdx=cos[(2a−b)x]4(2a−b)−cosbx2b−cos[(2a+b)x]4(2a+b)
(83)
∫cos2axsinaxdx=−13acos3ax
∫cos2axsinaxdx=−13acos3ax
(84)
∫sin2axcos2bxdx=x4−sin2ax8a−sin[2(a−b)x]16(a−b)+sin2bx8b−sin[2(a+b)x]16(a+b)
∫sin2axcos2bxdx=x4−sin2ax8a−sin[2(a−b)x]16(a−b)+sin2bx8b−sin[2(a+b)x]16(a+b)
(85)
∫sin2axcos2axdx=x8−sin4ax32a
∫sin2axcos2axdx=x8−sin4ax32a
(86)
∫tanaxdx=−1alncosax
∫tanaxdx=−1alncosax
(87)
∫tan2axdx=−x+1atanax
∫tan2axdx=−x+1atanax
(88)
∫tannaxdx=tann+1axa(1+n)×2F1(n+12,1,n+32,−tan2ax)
∫tannaxdx=tann+1axa(1+n)×2F1n+12,1,n+32,−tan2ax
(89)
∫tan3axdx=1alncosax+12asec2ax
∫tan3axdx=1alncosax+12asec2ax
(90)
∫secxdx=ln∣∣∣secx+tanx∣∣∣=2tanh−1(tanx2)
∫secxdx=ln|secx+tanx|=2tanh−1tanx2
(91)
∫sec2axdx=1atanax
∫sec2axdx=1atanax
(92)
∫sec3xdx=12secxtanx+12ln∣∣∣secx+tanx∣∣∣
∫sec3xdx=12secxtanx+12ln|secx+tanx|
(93)
∫secxtanxdx=secx
∫secxtanxdx=secx
(94)
∫sec2xtanxdx=12sec2x
∫sec2xtanxdx=12sec2x
(95)
∫secnxtanxdx=1nsecnx,n≠0
∫secnxtanxdx=1nsecnx,n≠0
(96)
∫cscxdx=ln∣∣tanx2∣∣=ln∣∣∣cscx−cotx∣∣∣+C
∫cscxdx=lntanx2=ln|cscx−cotx|+C
(97)
∫csc2axdx=−1acotax
∫csc2axdx=−1acotax
(98)
∫csc3xdx=−12cotxcscx+12ln∣∣∣cscx−cotx∣∣∣
∫csc3xdx=−12cotxcscx+12ln|cscx−cotx|
(99)
∫cscnxcotxdx=−1ncscnx,n≠0
∫cscnxcotxdx=−1ncscnx,n≠0
(100)
∫secxcscxdx=ln∣∣∣tanx∣∣∣
∫secxcscxdx=ln|tanx|
(101)
Products of Trigonometric Functions and Monomials
∫xcosxdx=cosx+xsinx
∫xcosxdx=cosx+xsinx
(102)
∫xcosaxdx=1a2cosax+xasinax
∫xcosaxdx=1a2cosax+xasinax
(103)
∫x2cosxdx=2xcosx+(x2−2)sinx
∫x2cosxdx=2xcosx+x2−2sinx
(104)
∫x2cosaxdx=2xcosaxa2+a2x2−2a3sinax
∫x2cosaxdx=2xcosaxa2+a2x2−2a3sinax
(105)
∫xncosxdx=−12(i)n+1[Γ(n+1,−ix)+(−1)nΓ(n+1,ix)]
∫xncosxdx=−12(i)n+1Γ(n+1,−ix)+(−1)nΓ(n+1,ix)
(106)
∫xncosaxdx=12(ia)1−n[(−1)nΓ(n+1,−iax)−Γ(n+1,ixa)]
∫xncosaxdx=12(ia)1−n(−1)nΓ(n+1,−iax)−Γ(n+1,ixa)
(107)
∫xsinxdx=−xcosx+sinx
∫xsinxdx=−xcosx+sinx
(108)
∫xsinaxdx=−xcosaxa+sinaxa2
∫xsinaxdx=−xcosaxa+sinaxa2
(109)
∫x2sinxdx=(2−x2)cosx+2xsinx
∫x2sinxdx=2−x2cosx+2xsinx
(110)
∫x2sinaxdx=2−a2x2a3cosax+2xsinaxa2
∫x2sinaxdx=2−a2x2a3cosax+2xsinaxa2
(111)
∫xnsinxdx=−12(i)n[Γ(n+1,−ix)−(−1)nΓ(n+1,−ix)]
∫xnsinxdx=−12(i)nΓ(n+1,−ix)−(−1)nΓ(n+1,−ix)
(112)
∫xcos2xdx=x24+18cos2x+14xsin2x
∫xcos2xdx=x24+18cos2x+14xsin2x
(113)
∫xsin2xdx=x24−18cos2x−14xsin2x
∫xsin2xdx=x24−18cos2x−14xsin2x
(114)
∫xtan2xdx=−x22+lncosx+xtanx
∫xtan2xdx=−x22+lncosx+xtanx
(115)
∫xsec2xdx=lncosx+xtanx
∫xsec2xdx=lncosx+xtanx
(116)
Products of Trigonometric Functions and Exponentials
∫exsinxdx=12ex(sinx−cosx)
∫exsinxdx=12ex(sinx−cosx)
(117)
∫ebxsinaxdx=1a2+b2ebx(bsinax−acosax)
∫ebxsinaxdx=1a2+b2ebx(bsinax−acosax)
(118)
∫excosxdx=12ex(sinx+cosx)
∫excosxdx=12ex(sinx+cosx)
(119)
∫ebxcosaxdx=1a2+b2ebx(asinax+bcosax)
∫ebxcosaxdx=1a2+b2ebx(asinax+bcosax)
(120)
∫xexsinxdx=12ex(cosx−xcosx+xsinx)
∫xexsinxdx=12ex(cosx−xcosx+xsinx)
(121)
∫xexcosxdx=12ex(xcosx−sinx+xsinx)
∫xexcosxdx=12ex(xcosx−sinx+xsinx)
(122)
Integrals of Hyperbolic Functions
∫coshaxdx=1asinhax
∫coshaxdx=1asinhax
(123)
∫eaxcoshbxdx={eaxa2−b2[acoshbx−bsinhbx]e2ax4a+x2a≠ba=b
∫eaxcoshbxdx=eaxa2−b2[acoshbx−bsinhbx]a≠be2ax4a+x2a=b
(124)
∫sinhaxdx=1acoshax
∫sinhaxdx=1acoshax
(125)
∫eaxsinhbxdx={eaxa2−b2[−bcoshbx+asinhbx]e2ax4a−x2a≠ba=b
∫eaxsinhbxdx=eaxa2−b2[−bcoshbx+asinhbx]a≠be2ax4a−x2a=b
(126)
∫tanhaxdx=1alncoshax
∫tanhaxdx=1alncoshax
(127)
∫eaxtanhbxdx=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪e(a+2b)x(a+2b)(2F1)[1+a2b,1,2+a2b,−e2bx]−eaxa(2F1)[1,a2b,1+a2b,−e2bx]eax−2tan−1[eax]aa≠ba=b
∫eaxtanhbxdx=e(a+2b)x(a+2b)(2F1)1+a2b,1,2+a2b,−e2bx−eaxa(2F1)1,a2b,1+a2b,−e2bxa≠beax−2tan−1[eax]aa=b
(128)
∫cosaxcoshbxdx=1a2+b2[asinaxcoshbx+bcosaxsinhbx]
∫cosaxcoshbxdx=1a2+b2asinaxcoshbx+bcosaxsinhbx
(129)
∫cosaxsinhbxdx=1a2+b2[bcosaxcoshbx+asinaxsinhbx]
∫cosaxsinhbxdx=1a2+b2bcosaxcoshbx+asinaxsinhbx
(130)
∫sinaxcoshbxdx=1a2+b2[−acosaxcoshbx+bsinaxsinhbx]
∫sinaxcoshbxdx=1a2+b2−acosaxcoshbx+bsinaxsinhbx
(131)
∫sinaxsinhbxdx=1a2+b2[bcoshbxsinax−acosaxsinhbx]
∫sinaxsinhbxdx=1a2+b2bcoshbxsinax−acosaxsinhbx
(132)
∫sinhaxcoshaxdx=14a[−2ax+sinh2ax]
∫sinhaxcoshaxdx=14a−2ax+sinh2ax
(133)
∫sinhaxcoshbxdx=1b2−a2[bcoshbxsinhax−acoshaxsinhbx]
∫sinhaxcoshbxdx=1b2−a2bcoshbxsinhax−acoshaxsinhbx
(134)
Basic Forms
∫xndx=1n+1xn+1,n≠−1
∫xndx=1n+1xn+1,n≠−1
(1)
∫1xdx=ln∣∣∣x∣∣∣
∫1xdx=ln|x|
(2)
∫udv=uv−∫vdu
∫udv=uv−∫vdu
(3)
∫1ax+bdx=1aln∣∣∣ax+b∣∣∣
∫1ax+bdx=1aln|ax+b|
(4)
Integrals of Rational Functions
∫1(x+a)2dx=−1x+a
∫1(x+a)2dx=−1x+a
(5)
∫(x+a)ndx=(x+a)n+1n+1,n≠−1
∫(x+a)ndx=(x+a)n+1n+1,n≠−1
(6)
∫x(x+a)ndx=(x+a)n+1((n+1)x−a)(n+1)(n+2)
∫x(x+a)ndx=(x+a)n+1((n+1)x−a)(n+1)(n+2)
(7)
∫11+x2dx=tan−1x
∫11+x2dx=tan−1x
(8)
∫1a2+x2dx=1atan−1xa
∫1a2+x2dx=1atan−1xa
(9)
∫xa2+x2dx=12ln∣∣∣a2+x2∣∣∣
∫xa2+x2dx=12ln|a2+x2|
(10)
∫x2a2+x2dx=x−atan−1xa
∫x2a2+x2dx=x−atan−1xa
(11)
∫x3a2+x2dx=12x2−12a2ln∣∣∣a2+x2∣∣∣
∫x3a2+x2dx=12x2−12a2ln|a2+x2|
(12)
∫1ax2+bx+cdx=24ac−b2−−−−−−−√tan−12ax+b4ac−b2−−−−−−−√
∫1ax2+bx+cdx=24ac−b2tan−12ax+b4ac−b2
(13)
∫1(x+a)(x+b)dx=1b−alna+xb+x, a≠b
∫1(x+a)(x+b)dx=1b−alna+xb+x, a≠b
(14)
∫x(x+a)2dx=aa+x+ln∣∣∣a+x∣∣∣
∫x(x+a)2dx=aa+x+ln|a+x|
(15)
∫xax2+bx+cdx=12aln∣∣∣ax2+bx+c∣∣∣−ba4ac−b2−−−−−−−√tan−12ax+b4ac−b2−−−−−−−√
∫xax2+bx+cdx=12aln|ax2+bx+c|−ba4ac−b2tan−12ax+b4ac−b2
(16)
Integrals with Roots
∫x−a−−−−−√dx=23(x−a)3∕2
∫x−adx=23(x−a)3∕2
(17)
∫1x±a−−−−−√dx=2x±a−−−−−√
∫1x±adx=2x±a
(18)
∫1a−x−−−−−√dx=−2a−x−−−−−√
∫1a−xdx=−2a−x
(19)
∫xx−a−−−−−√dx=⎧⎩⎨⎪⎪⎪⎪⎪⎪2a3(x−a)3∕2+25(x−a)5∕2, or23x(x−a)3∕2−415(x−a)5∕2, or215(2a+3x)(x−a)3∕2
∫xx−adx=2a3x−a3∕2+25x−a5∕2, or23x(x−a)3∕2−415(x−a)5∕2, or215(2a+3x)(x−a)3∕2
(20)
∫ax+b−−−−−√dx=(2b3a+2x3)ax+b−−−−−√
∫ax+bdx=2b3a+2x3ax+b
(21)
∫(ax+b)3∕2dx=25a(ax+b)5∕2
∫(ax+b)3∕2dx=25a(ax+b)5∕2
(22)
∫xx±a−−−−−√dx=23(x∓2a)x±a−−−−−√
∫xx±adx=23(x∓2a)x±a
(23)
∫xa−x−−−−−√dx=−x(a−x)−−−−−−−√−atan−1x(a−x)−−−−−−−√x−a
∫xa−xdx=−x(a−x)−atan−1x(a−x)x−a
(24)
∫xa+x−−−−−√dx=x(a+x)−−−−−−−√−aln[x−−√+x+a−−−−−√]
∫xa+xdx=x(a+x)−alnx+x+a
(25)
∫xax+b−−−−−√dx=215a2(−2b2+abx+3a2x2)ax+b−−−−−√
∫xax+bdx=215a2(−2b2+abx+3a2x2)ax+b
(26)
∫x(ax+b)−−−−−−−−√dx=14a3∕2[(2ax+b)ax(ax+b)−−−−−−−−−√−b2ln∣∣ax−−√+a(ax+b)−−−−−−−−√∣∣]
∫x(ax+b)dx=14a3∕2(2ax+b)ax(ax+b)−b2lnax+a(ax+b)
(27)
∫x3(ax+b)−−−−−−−−−√dx=[b12a−b28a2x+x3]x3(ax+b)−−−−−−−−−√+b38a5∕2ln∣∣ax−−√+a(ax+b)−−−−−−−−√∣∣
∫x3(ax+b)dx=b12a−b28a2x+x3x3(ax+b)+b38a5∕2lnax+a(ax+b)
(28)
∫x2±a2−−−−−−√dx=12xx2±a2−−−−−−√±12a2ln∣∣x+x2±a2−−−−−−√∣∣
∫x2±a2dx=12xx2±a2±12a2lnx+x2±a2
(29)
∫a2−x2−−−−−−√dx=12xa2−x2−−−−−−√+12a2tan−1xa2−x2−−−−−−√
∫a2−x2dx=12xa2−x2+12a2tan−1xa2−x2
(30)
∫xx2±a2−−−−−−√dx=13(x2±a2)3∕2
∫xx2±a2dx=13x2±a23∕2
(31)
∫1x2±a2−−−−−−√dx=ln∣∣x+x2±a2−−−−−−√∣∣
∫1x2±a2dx=lnx+x2±a2
(32)
∫1a2−x2−−−−−−√dx=sin−1xa
∫1a2−x2dx=sin−1xa
(33)
∫xx2±a2−−−−−−√dx=x2±a2−−−−−−√
∫xx2±a2dx=x2±a2
(34)
∫xa2−x2−−−−−−√dx=−a2−x2−−−−−−√
∫xa2−x2dx=−a2−x2
(35)
∫x2x2±a2−−−−−−√dx=12xx2±a2−−−−−−√∓12a2ln∣∣x+x2±a2−−−−−−√∣∣
∫x2x2±a2dx=12xx2±a2∓12a2lnx+x2±a2
(36)
∫ax2+bx+c−−−−−−−−−−√dx=b+2ax4aax2+bx+c−−−−−−−−−−√+4ac−b28a3∕2ln∣∣2ax+b+2a(ax2+bx+c)−−−−−−−−−−−−√∣∣
∫ax2+bx+cdx=b+2ax4aax2+bx+c+4ac−b28a3∕2ln2ax+b+2a(ax2+bx+c)
(37)
∫xax2+bx+c−−−−−−−−−−√dx=148a5∕2(2a−−√ax2+bx+c−−−−−−−−−−√(−3b2+2abx+8a(c+ax2))+3(b3−4abc)ln∣∣b+2ax+2a−−√ax2+bx+c−−−−−−−−−−√∣∣)
∫xax2+bx+cdx=148a5∕22aax2+bx+c−3b2+2abx+8a(c+ax2)+3(b3−4abc)lnb+2ax+2aax2+bx+c
(38)
∫1ax2+bx+c−−−−−−−−−−√dx=1a−−√ln∣∣2ax+b+2a(ax2+bx+c)−−−−−−−−−−−−−√∣∣
∫1ax2+bx+cdx=1aln2ax+b+2a(ax2+bx+c)
(39)
∫xax2+bx+c−−−−−−−−−−√dx=1aax2+bx+c−−−−−−−−−−√−b2a3∕2ln∣∣2ax+b+2a(ax2+bx+c)−−−−−−−−−−−−−√∣∣
∫xax2+bx+cdx=1aax2+bx+c−b2a3∕2ln2ax+b+2a(ax2+bx+c)
(40)
∫dx(a2+x2)3∕2=xa2a2+x2−−−−−−√
∫dx(a2+x2)3∕2=xa2a2+x2
(41)
Integrals with Logarithms
∫lnaxdx=xlnax−x
∫lnaxdx=xlnax−x
(42)
∫xlnxdx=12x2lnx−x24
∫xlnxdx=12x2lnx−x24
(43)
∫x2lnxdx=13x3lnx−x39
∫x2lnxdx=13x3lnx−x39
(44)
∫xnlnxdx=xn+1(lnxn+1−1(n+1)2),n≠−1
∫xnlnxdx=xn+1lnxn+1−1(n+1)2,n≠−1
(45)
∫lnaxxdx=12(lnax)2
∫lnaxxdx=12lnax2
(46)
∫lnxx2dx=−1x−lnxx
∫lnxx2dx=−1x−lnxx
(47)
∫ln(ax+b)dx=(x+ba)ln(ax+b)−x,a≠0
∫ln(ax+b)dx=x+baln(ax+b)−x,a≠0
(48)
∫ln(x2+a2)dx=xln(x2+a2)+2atan−1xa−2x
∫ln(x2+a2)dx=xln(x2+a2)+2atan−1xa−2x
(49)
∫ln(x2−a2)dx=xln(x2−a2)+alnx+ax−a−2x
∫ln(x2−a2)dx=xln(x2−a2)+alnx+ax−a−2x
(50)
∫ln(ax2+bx+c)dx=1a4ac−b2−−−−−−−√tan−12ax+b4ac−b2−−−−−−−√−2x+(b2a+x)ln(ax2+bx+c)
∫lnax2+bx+cdx=1a4ac−b2tan−12ax+b4ac−b2−2x+b2a+xlnax2+bx+c
(51)
∫xln(ax+b)dx=bx2a−14x2+12(x2−b2a2)ln(ax+b)
∫xln(ax+b)dx=bx2a−14x2+12x2−b2a2ln(ax+b)
(52)
∫xln(a2−b2x2)dx=−12x2+12(x2−a2b2)ln(a2−b2x2)
∫xlna2−b2x2dx=−12x2+12x2−a2b2lna2−b2x2
(53)
∫(lnx)2dx=2x−2xlnx+x(lnx)2
∫(lnx)2dx=2x−2xlnx+x(lnx)2
(54)
∫(lnx)3dx=−6x+x(lnx)3−3x(lnx)2+6xlnx
∫(lnx)3dx=−6x+x(lnx)3−3x(lnx)2+6xlnx
(55)
∫x(lnx)2dx=x24+12x2(lnx)2−12x2lnx
∫x(lnx)2dx=x24+12x2(lnx)2−12x2lnx
(56)
∫x2(lnx)2dx=2x327+13x3(lnx)2−29x3lnx
∫x2(lnx)2dx=2x327+13x3(lnx)2−29x3lnx
(57)
Integrals with Exponentials
∫eaxdx=1aeax
∫eaxdx=1aeax
(58)
∫x−−√eaxdx=1ax−−√eax+iπ−−√2a3∕2erf(iax−−√), where erf(x)=2π−−√∫x0e−t2dt
∫xeaxdx=1axeax+iπ2a3∕2erfiax, where erf(x)=2π∫0xe−t2dt
(59)
∫xexdx=(x−1)ex
∫xexdx=(x−1)ex
(60)
∫xeaxdx=(xa−1a2)eax
∫xeaxdx=xa−1a2eax
(61)
∫x2exdx=(x2−2x+2)ex
∫x2exdx=x2−2x+2ex
(62)
∫x2eaxdx=(x2a−2xa2+2a3)eax
∫x2eaxdx=x2a−2xa2+2a3eax
(63)
∫x3exdx=(x3−3x2+6x−6)ex
∫x3exdx=x3−3x2+6x−6ex
(64)
∫xneaxdx=xneaxa−na∫xn−1eaxdx
∫xneaxdx=xneaxa−na∫xn−1eaxdx
(65)
∫xneaxdx=(−1)nan+1Γ[1+n,−ax], where Γ(a,x)=∫∞xta−1e−tdt
∫xneaxdx=(−1)nan+1Γ[1+n,−ax], where Γ(a,x)=∫x∞ta−1e−tdt
(66)
∫eax2dx=−iπ−−√2a−−√erf(ixa−−√)
∫eax2dx=−iπ2aerfixa
(67)
∫e−ax2dx=π−−√2a−−√erf(xa−−√)
∫e−ax2dx=π2aerfxa
(68)
∫xe−ax2dx=−12ae−ax2
∫xe−ax2dx=−12ae−ax2
(69)
∫x2e−ax2dx=14πa3−−−√erf(xa−−√)−x2ae−ax2
∫x2e−ax2dx=14πa3erf(xa)−x2ae−ax2
(70)
Integrals with Trigonometric Functions
∫sinaxdx=−1acosax
∫sinaxdx=−1acosax
(71)
∫sin2axdx=x2−sin2ax4a
∫sin2axdx=x2−sin2ax4a
(72)
∫sin3axdx=−3cosax4a+cos3ax12a
∫sin3axdx=−3cosax4a+cos3ax12a
(73)
∫sinnaxdx=−1acosax2F1[12,1−n2,32,cos2ax]
∫sinnaxdx=−1acosax2F112,1−n2,32,cos2ax
(74)
∫cosaxdx=1asinax
∫cosaxdx=1asinax
(75)
∫cos2axdx=x2+sin2ax4a
∫cos2axdx=x2+sin2ax4a
(76)
∫cos3axdx=3sinax4a+sin3ax12a
∫cos3axdx=3sinax4a+sin3ax12a
(77)
∫cospaxdx=−1a(1+p)cos1+pax×2F1[1+p2,12,3+p2,cos2ax]
∫cospaxdx=−1a(1+p)cos1+pax×2F11+p2,12,3+p2,cos2ax
(78)
∫cosxsinxdx=12sin2x+c1=−12cos2x+c2=−14cos2x+c3
∫cosxsinxdx=12sin2x+c1=−12cos2x+c2=−14cos2x+c3
(79)
∫cosaxsinbxdx=cos[(a−b)x]2(a−b)−cos[(a+b)x]2(a+b),a≠b
∫cosaxsinbxdx=cos[(a−b)x]2(a−b)−cos[(a+b)x]2(a+b),a≠b
(80)
∫sin2axcosbxdx=−sin[(2a−b)x]4(2a−b)+sinbx2b−sin[(2a+b)x]4(2a+b)
∫sin2axcosbxdx=−sin[(2a−b)x]4(2a−b)+sinbx2b−sin[(2a+b)x]4(2a+b)
(81)
∫sin2xcosxdx=13sin3x
∫sin2xcosxdx=13sin3x
(82)
∫cos2axsinbxdx=cos[(2a−b)x]4(2a−b)−cosbx2b−cos[(2a+b)x]4(2a+b)
∫cos2axsinbxdx=cos[(2a−b)x]4(2a−b)−cosbx2b−cos[(2a+b)x]4(2a+b)
(83)
∫cos2axsinaxdx=−13acos3ax
∫cos2axsinaxdx=−13acos3ax
(84)
∫sin2axcos2bxdx=x4−sin2ax8a−sin[2(a−b)x]16(a−b)+sin2bx8b−sin[2(a+b)x]16(a+b)
∫sin2axcos2bxdx=x4−sin2ax8a−sin[2(a−b)x]16(a−b)+sin2bx8b−sin[2(a+b)x]16(a+b)
(85)
∫sin2axcos2axdx=x8−sin4ax32a
∫sin2axcos2axdx=x8−sin4ax32a
(86)
∫tanaxdx=−1alncosax
∫tanaxdx=−1alncosax
(87)
∫tan2axdx=−x+1atanax
∫tan2axdx=−x+1atanax
(88)
∫tannaxdx=tann+1axa(1+n)×2F1(n+12,1,n+32,−tan2ax)
∫tannaxdx=tann+1axa(1+n)×2F1n+12,1,n+32,−tan2ax
(89)
∫tan3axdx=1alncosax+12asec2ax
∫tan3axdx=1alncosax+12asec2ax
(90)
∫secxdx=ln∣∣∣secx+tanx∣∣∣=2tanh−1(tanx2)
∫secxdx=ln|secx+tanx|=2tanh−1tanx2
(91)
∫sec2axdx=1atanax
∫sec2axdx=1atanax
(92)
∫sec3xdx=12secxtanx+12ln∣∣∣secx+tanx∣∣∣
∫sec3xdx=12secxtanx+12ln|secx+tanx|
(93)
∫secxtanxdx=secx
∫secxtanxdx=secx
(94)
∫sec2xtanxdx=12sec2x
∫sec2xtanxdx=12sec2x
(95)
∫secnxtanxdx=1nsecnx,n≠0
∫secnxtanxdx=1nsecnx,n≠0
(96)
∫cscxdx=ln∣∣tanx2∣∣=ln∣∣∣cscx−cotx∣∣∣+C
∫cscxdx=lntanx2=ln|cscx−cotx|+C
(97)
∫csc2axdx=−1acotax
∫csc2axdx=−1acotax
(98)
∫csc3xdx=−12cotxcscx+12ln∣∣∣cscx−cotx∣∣∣
∫csc3xdx=−12cotxcscx+12ln|cscx−cotx|
(99)
∫cscnxcotxdx=−1ncscnx,n≠0
∫cscnxcotxdx=−1ncscnx,n≠0
(100)
∫secxcscxdx=ln∣∣∣tanx∣∣∣
∫secxcscxdx=ln|tanx|
(101)
Products of Trigonometric Functions and Monomials
∫xcosxdx=cosx+xsinx
∫xcosxdx=cosx+xsinx
(102)
∫xcosaxdx=1a2cosax+xasinax
∫xcosaxdx=1a2cosax+xasinax
(103)
∫x2cosxdx=2xcosx+(x2−2)sinx
∫x2cosxdx=2xcosx+x2−2sinx
(104)
∫x2cosaxdx=2xcosaxa2+a2x2−2a3sinax
∫x2cosaxdx=2xcosaxa2+a2x2−2a3sinax
(105)
∫xncosxdx=−12(i)n+1[Γ(n+1,−ix)+(−1)nΓ(n+1,ix)]
∫xncosxdx=−12(i)n+1Γ(n+1,−ix)+(−1)nΓ(n+1,ix)
(106)
∫xncosaxdx=12(ia)1−n[(−1)nΓ(n+1,−iax)−Γ(n+1,ixa)]
∫xncosaxdx=12(ia)1−n(−1)nΓ(n+1,−iax)−Γ(n+1,ixa)
(107)
∫xsinxdx=−xcosx+sinx
∫xsinxdx=−xcosx+sinx
(108)
∫xsinaxdx=−xcosaxa+sinaxa2
∫xsinaxdx=−xcosaxa+sinaxa2
(109)
∫x2sinxdx=(2−x2)cosx+2xsinx
∫x2sinxdx=2−x2cosx+2xsinx
(110)
∫x2sinaxdx=2−a2x2a3cosax+2xsinaxa2
∫x2sinaxdx=2−a2x2a3cosax+2xsinaxa2
(111)
∫xnsinxdx=−12(i)n[Γ(n+1,−ix)−(−1)nΓ(n+1,−ix)]
∫xnsinxdx=−12(i)nΓ(n+1,−ix)−(−1)nΓ(n+1,−ix)
(112)
∫xcos2xdx=x24+18cos2x+14xsin2x
∫xcos2xdx=x24+18cos2x+14xsin2x
(113)
∫xsin2xdx=x24−18cos2x−14xsin2x
∫xsin2xdx=x24−18cos2x−14xsin2x
(114)
∫xtan2xdx=−x22+lncosx+xtanx
∫xtan2xdx=−x22+lncosx+xtanx
(115)
∫xsec2xdx=lncosx+xtanx
∫xsec2xdx=lncosx+xtanx
(116)
Products of Trigonometric Functions and Exponentials
∫exsinxdx=12ex(sinx−cosx)
∫exsinxdx=12ex(sinx−cosx)
(117)
∫ebxsinaxdx=1a2+b2ebx(bsinax−acosax)
∫ebxsinaxdx=1a2+b2ebx(bsinax−acosax)
(118)
∫excosxdx=12ex(sinx+cosx)
∫excosxdx=12ex(sinx+cosx)
(119)
∫ebxcosaxdx=1a2+b2ebx(asinax+bcosax)
∫ebxcosaxdx=1a2+b2ebx(asinax+bcosax)
(120)
∫xexsinxdx=12ex(cosx−xcosx+xsinx)
∫xexsinxdx=12ex(cosx−xcosx+xsinx)
(121)
∫xexcosxdx=12ex(xcosx−sinx+xsinx)
∫xexcosxdx=12ex(xcosx−sinx+xsinx)
(122)
Integrals of Hyperbolic Functions
∫coshaxdx=1asinhax
∫coshaxdx=1asinhax
(123)
∫eaxcoshbxdx={eaxa2−b2[acoshbx−bsinhbx]e2ax4a+x2a≠ba=b
∫eaxcoshbxdx=eaxa2−b2[acoshbx−bsinhbx]a≠be2ax4a+x2a=b
(124)
∫sinhaxdx=1acoshax
∫sinhaxdx=1acoshax
(125)
∫eaxsinhbxdx={eaxa2−b2[−bcoshbx+asinhbx]e2ax4a−x2a≠ba=b
∫eaxsinhbxdx=eaxa2−b2[−bcoshbx+asinhbx]a≠be2ax4a−x2a=b
(126)
∫tanhaxdx=1alncoshax
∫tanhaxdx=1alncoshax
(127)
∫eaxtanhbxdx=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪e(a+2b)x(a+2b)(2F1)[1+a2b,1,2+a2b,−e2bx]−eaxa(2F1)[1,a2b,1+a2b,−e2bx]eax−2tan−1[eax]aa≠ba=b
∫eaxtanhbxdx=e(a+2b)x(a+2b)(2F1)1+a2b,1,2+a2b,−e2bx−eaxa(2F1)1,a2b,1+a2b,−e2bxa≠beax−2tan−1[eax]aa=b
(128)
∫cosaxcoshbxdx=1a2+b2[asinaxcoshbx+bcosaxsinhbx]
∫cosaxcoshbxdx=1a2+b2asinaxcoshbx+bcosaxsinhbx
(129)
∫cosaxsinhbxdx=1a2+b2[bcosaxcoshbx+asinaxsinhbx]
∫cosaxsinhbxdx=1a2+b2bcosaxcoshbx+asinaxsinhbx
(130)
∫sinaxcoshbxdx=1a2+b2[−acosaxcoshbx+bsinaxsinhbx]
∫sinaxcoshbxdx=1a2+b2−acosaxcoshbx+bsinaxsinhbx
(131)
∫sinaxsinhbxdx=1a2+b2[bcoshbxsinax−acosaxsinhbx]
∫sinaxsinhbxdx=1a2+b2bcoshbxsinax−acosaxsinhbx
(132)
∫sinhaxcoshaxdx=14a[−2ax+sinh2ax]
∫sinhaxcoshaxdx=14a−2ax+sinh2ax
(133)
∫sinhaxcoshbxdx=1b2−a2[bcoshbxsinhax−acoshaxsinhbx]
∫sinhaxcoshbxdx=1b2−a2bcoshbxsinhax−acoshaxsinhbx
(134)
Comments
Post a Comment