integral-table.pdf

Table of Basic Integrals Basic Forms

(1)

Z

xⁿdx= 1

n+ 1xⁿ⁺¹, n6=−1

(2)

Z 1

xdx= ln|x|

(3)

Z

udv=uv − Z

vdu

(4)

Z 1

ax+bdx= 1

aln|ax+b|

Integrals of Rational Functions

(5)

Z 1

(x+a)²dx=− 1 x+a

(6)

Z

(x+a)ⁿdx= (x+a)ⁿ⁺¹

n+ 1 , n6=−1

(7)

Z

x(x+a)ⁿdx= (x+a)ⁿ⁺¹((n+ 1)x−a) (n+ 1)(n+ 2)

(8)

Z 1

1 +x²dx= tan⁻¹x

(9)

Z 1

a²+x²dx= 1

atan⁻¹ x a

(10)

Z x

a²+x²dx= 1

2ln|a²+x²|

(11)

Z x²

a² +x²dx=x−atan⁻¹ x a

(12)

Z x³

a²+x²dx= 1

2x²− 1

2a²ln|a²+x²|

(13)

Z 1

ax² +bx+cdx= 2

√4ac−b² tan⁻¹ 2ax+b

√4ac−b²

(14)

Z 1

(x+a)(x+b)dx= 1

b−alna+x

b+x, a6=b

(15)

Z x

(x+a)²dx= a

a+x + ln|a+x|

(16)

Z x

ax²+bx+cdx= 1

2aln|ax²+bx+c|− b a√

4ac−b² tan⁻¹ 2ax+b

√4ac−b²

Integrals with Roots

(17)

Z √

x−a dx= 2

3(x−a)^3/2

(18)

Z 1

√x±a dx= 2√ x±a

(19)

Z 1

√a−x dx=−2√ a−x

(20)

Z x√

x−a dx=







2a

3 (x−a)^3/2+²₅ (x−a)^5/2, or

2

3x(x−a)^3/2− ₁₅⁴(x−a)^5/2, or

2

15(2a+ 3x)(x−a)^3/2 (21)

Z √

ax+b dx = 2b

3a + 2x 3

√ ax+b

(22)

Z

(ax+b)^3/2 dx= 2

5a(ax+b)^5/2

(23)

Z x

√x±a dx= 2

3(x∓2a)√ x±a

(24)

Z r x

a−x dx=−p

x(a−x)−atan⁻¹

px(a−x) x−a

(25)

Z r x

a+x dx=p

x(a+x)−aln√ x+√

x+a

(26)

Z x√

ax+b dx= 2

15a²(−2b²+abx+ 3a²x²)√ ax+b

(27) Z

px(ax+b)dx= 1 4a^3/2

h

(2ax+b)p

ax(ax+b)−b²ln a√

x+p

a(ax+b) i

(28) Z

px³(ax+b)dx= b

12a − b² 8a²x + x

3

px³(ax+b)+ b³ 8a^5/2 ln

a√

x+p

a(ax+b)

(29)

Z √

x²±a² dx= 1 2x√

x²±a²± 1 2a²ln

x+√

x²±a²

(30)

Z √

a²−x² dx= 1 2x√

a²−x²+ 1

2a²tan⁻¹ x

√a²−x²

(31)

Z x√

x²±a² dx= 1

3 x²±a²3/2

(32)

Z 1

√x² ±a² dx= ln x+√

x²±a²

(33)

Z 1

√a²−x² dx= sin⁻¹ x a

(34)

Z x

√x²±a² dx=√

x²±a²

(35)

Z x

√a²−x² dx=−√

a²−x²

(36)

Z x²

√x²±a² dx= 1 2x√

x²±a²∓1 2a²ln

x+√

x²±a²

(37) Z √

ax²+bx+c dx= b+ 2ax 4a

√

ax²+bx+c+4ac−b² 8a^3/2 ln

2ax+b+ 2p

a(ax²+bx⁺c)

Z x√

ax²+bx+c dx= 1 48a^5/2

2√

a√

ax²+bx+c −3b²+ 2abx+ 8a(c+ax²)

+3(b³−4abc) ln

b+ 2ax+ 2√ a√

ax²+bx+c (38)

(39)

Z 1

√ax²+bx+c dx= 1

√aln

2ax+b+ 2p

a(ax² +bx+c)

(40)

Z x

√ax²+bx+cdx= 1 a

√

ax²+bx+c− b 2a^3/2 ln

2ax+b+ 2p

a(ax²+bx+c)

(41)

Z dx

(a²+x²)^3/2 = x a²√

a²+x²

Integrals with Logarithms

(42)

Z

lnax dx=xlnax−x

(43)

Z

xlnx dx= 1

2x²lnx− x² 4

(44)

Z

x²lnx dx= 1

3x³lnx− x³ 9

(45)

Z

xⁿlnx dx=xⁿ⁺¹

lnx

n+ 1 − 1 (n+ 1)²

, n6=−1

(46)

Z lnax

x dx= 1

2(lnax)²

(47)

Z lnx

x² dx =−1

x −lnx x

(48)

Z

ln(ax+b) dx=

x+ b a

ln(ax+b)−x, a6= 0

(49)

Z

ln(x²+a²)dx=xln(x²+a²) + 2atan⁻¹ x a −2x

(50)

Z

ln(x²−a²)dx=xln(x² −a²) +alnx+a x−a −2x

(51) Z

ln ax²+bx+c

dx= 1 a

√

4ac−b²tan⁻¹ 2ax+b

√4ac−b²−2x+

b 2a +x

ln ax²+bx+c

(52)

Z

xln(ax+b) dx= bx 2a − 1

4x²+ 1 2

x²− b² a²

ln(ax+b)

(53)

Z

xln a²−b²x²

dx =−1

2x² +1 2

x²− a² b²

ln a²−b²x²

(54)

Z

(lnx)² dx= 2x−2xlnx+x(lnx)²

(55)

Z

(lnx)³ dx=−6x+x(lnx)³−3x(lnx)²+ 6xlnx

(56)

Z

x(lnx)² dx= x² 4 +1

2x²(lnx)²− 1 2x²lnx

(57)

Z

x²(lnx)² dx= 2x³ 27 +1

3x³(lnx)²− 2 9x³lnx

Integrals with Exponentials

(58)

Z

e^ax dx= 1 ae^ax

(59) Z √

xe^ax dx= 1 a

√xe^ax+ i√ π

2a^3/2erf i√ ax

, where erf(x) = 2

√π Z x

0

e^−t2dt

(60)

Z

xe^x dx= (x−1)e^x

(61)

Z

xe^ax dx= x

a − 1 a²

e^ax

(62)

Z

x²e^x dx= x²−2x+ 2 e^x

(63)

Z

x²e^ax dx= x²

a − 2x a² + 2

a³

e^ax

(64)

Z

x³e^x dx= x³−3x²+ 6x−6 e^x

(65)

Z

xⁿe^ax dx= xⁿe^ax a −n

a Z

xⁿ⁻¹e^axdx

(66) Z

xⁿe^ax dx= (−1)ⁿ

aⁿ⁺¹ Γ[1 +n,−ax], where Γ(a, x) = Z ∞

x

t^a−1e^−tdt

(67)

Z

e^ax2 dx=−i√ π 2√

aerf ix√ a

(68)

Z

e^−ax2 dx=

√π 2√

aerf x√ a

(69)

Z

xe^−ax2 dx =− 1 2ae^−ax2

(70)

Z

x²e^−ax2 dx= 1 4

rπ

a³erf(x√

a)− x 2ae^−ax2

Integrals with Trigonometric Functions

(71)

Z

sinax dx=−1 acosax

(72)

Z

sin²ax dx= x

2 − sin 2ax 4a

(73)

Z

sin³ax dx=−3 cosax

4a +cos 3ax 12a

(74)

Z

sinⁿax dx=−1

acosax ₂F₁ 1

2,1−n 2 ,3

2,cos²ax

(75)

Z

cosax dx= 1 asinax

(76)

Z

cos²ax dx= x

2 + sin 2ax 4a

(77)

Z

cos³axdx= 3 sinax

4a + sin 3ax 12a

(78) Z

cos^paxdx=− 1

a(1 +p)cos^1+pax×₂F₁

1 +p 2 ,1

2,3 +p

2 ,cos²ax

(79) Z

cosxsinx dx= 1

2sin²x+c₁ =−1

2cos²x+c₂ =−1

4cos 2x+c₃

(80)

Z

cosaxsinbx dx = cos[(a−b)x]

2(a−b) − cos[(a+b)x]

2(a+b) , a6=b

(81) Z

sin²axcosbx dx=−sin[(2a−b)x]

4(2a−b) +sinbx

2b − sin[(2a+b)x]

4(2a+b)

(82)

Z

sin²xcosx dx= 1 3sin³x

(83)

Z

cos²axsinbx dx= cos[(2a−b)x]

4(2a−b) − cosbx

2b − cos[(2a+b)x]

4(2a+b)

(84)

Z

cos²axsinax dx=− 1

3acos³ax

(85) Z

sin²axcos²bxdx= x

4−sin 2ax

8a −sin[2(a−b)x]

16(a−b) +sin 2bx

8b −sin[2(a+b)x]

16(a+b)

(86)

Z

sin²axcos²ax dx= x

8 − sin 4ax 32a

(87)

Z

tanax dx =−1

aln cosax

(88)

Z

tan²ax dx=−x+1

atanax

(89)

Z

tanⁿax dx= tanⁿ⁺¹ax a(1 +n) ×₂F₁

n+ 1

2 ,1,n+ 3

2 ,−tan²ax

(90)

Z

tan³axdx= 1

aln cosax+ 1

2asec²ax

(91)

Z

secx dx= ln|secx+ tanx|= 2 tanh⁻¹ tanx

2

(92)

Z

sec²ax dx = 1

atanax

(93)

Z

sec³x dx= 1

2secxtanx+1

2ln|secx+ tanx|

(94)

Z

secxtanx dx= secx

(95)

Z

sec²xtanx dx= 1 2sec²x

(96)

Z

secⁿxtanx dx= 1

n secⁿx, n6= 0

(97)

Z

cscx dx= ln tanx

2

= ln|cscx−cotx|+C

(98)

Z

csc²ax dx =−1 acotax

(99)

Z

csc³x dx=−1

2cotxcscx+1

2ln|cscx−cotx|

(100)

Z

cscⁿxcotx dx=−1

n cscⁿx, n6= 0

(101)

Z

secxcscx dx= ln|tanx|

Products of Trigonometric Functions and Mono- mials

(102)

Z

xcosx dx= cosx+xsinx

(103)

Z

xcosax dx= 1

a² cosax+x a sinax

(104)

Z

x²cosx dx= 2xcosx+ x² −2 sinx

(105)

Z

x²cosax dx= 2xcosax

a² + a²x²−2 a³ sinax

(106)

Z

xⁿcosxdx=−1

2(i)ⁿ⁺¹[Γ(n+ 1,−ix) + (−1)ⁿΓ(n+ 1, ix)]

(107) Z

xⁿcosax dx = 1

2(ia)¹⁻ⁿ[(−1)ⁿΓ(n+ 1,−iax)−Γ(n+ 1, ixa)]

(108)

Z

xsinx dx=−xcosx+ sinx

(109)

Z

xsinax dx=−xcosax

a +sinax a²

(110)

Z

x²sinx dx= 2−x²

cosx+ 2xsinx

(111)

Z

x²sinax dx= 2−a²x²

a³ cosax+2xsinax a²

(112) Z

xⁿsinx dx=−1

2(i)ⁿ[Γ(n+ 1,−ix)−(−1)ⁿΓ(n+ 1,−ix)]

(113)

Z

xcos²x dx= x² 4 + 1

8cos 2x+1

4xsin 2x

(114)

Z

xsin²x dx= x² 4 − 1

8cos 2x−1

4xsin 2x

(115)

Z

xtan²x dx=−x²

2 + ln cosx+xtanx

(116)

Z

xsec²x dx= ln cosx+xtanx

Products of Trigonometric Functions and Ex- ponentials

(117)

Z

e^xsinx dx= 1

2e^x(sinx−cosx)

(118)

Z

e^bxsinax dx= 1

a² +b²e^bx(bsinax−acosax)

(119)

Z

e^xcosx dx= 1

2e^x(sinx+ cosx)

(120)

Z

e^bxcosax dx= 1

a²+b²e^bx(asinax+bcosax)

(121)

Z

xe^xsinx dx= 1

2e^x(cosx−xcosx+xsinx)

(122)

Z

xe^xcosx dx= 1

2e^x(xcosx−sinx+xsinx)

Integrals of Hyperbolic Functions

(123)

Z

coshax dx= 1

asinhax

(124)

Z

e^axcoshbx dx=





 e^ax

a²−b²[acoshbx−bsinhbx] a 6=b e^2ax

4a + x

2 a =b

(125)

Z

sinhax dx= 1

acoshax

(126)

Z

e^axsinhbx dx=





 e^ax

a²−b²[−bcoshbx+asinhbx] a6=b e^2ax

4a −x

2 a=b

(127)

Z

tanhax dx= 1

aln coshax

(128) Z

e^axtanhbx dx=















e^(a+2b)x (a+ 2b)²F₁h

1 + a

2b,1,2 + a

2b,−e^2bxi

−1 ae^ax2F1

h 1, a

2b,1 + a

2b,−e^2bxi

a6=b e^ax−2 tan⁻¹[e^ax]

a a=b

(129) Z

cosaxcoshbx dx= 1

a²+b² [asinaxcoshbx+bcosaxsinhbx]

(130) Z

cosaxsinhbx dx = 1

a²+b² [bcosaxcoshbx+asinaxsinhbx]

(131) Z

sinaxcoshbx dx = 1

a²+b² [−acosaxcoshbx+bsinaxsinhbx]

(132) Z

sinaxsinhbx dx= 1

a²+b² [bcoshbxsinax−acosaxsinhbx]

(133)

Z

sinhaxcoshaxdx= 1

4a[−2ax+ sinh 2ax]

(134) Z

sinhaxcoshbx dx= 1

b²−a² [bcoshbxsinhax−acoshaxsinhbx]

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