Table of Basic Integrals Basic Forms

Table of Basic Integrals Basic Forms

Z

xⁿdx= 1

n+ 1xⁿ⁺¹, n6= −1 (1) Z 1

xdx= ln |x| (2)

Z

udv= uv − Z

vdu (3)

Z 1

ax+ bdx= 1

aln |ax + b| (4)

Integrals of Rational Functions

Z 1

(x + a)²dx= − 1

x+ a (5)

Z

(x + a)ⁿdx= (x + a)ⁿ⁺¹

n+ 1 , n6= −1 (6)

Z

x(x + a)ⁿdx= (x + a)ⁿ⁺¹((n + 1)x − a)

(n + 1)(n + 2) (7)

Z 1

1 + x²dx= tan⁻¹x (8)

Z 1

a²+ x²dx= 1

atan⁻¹ x

a (9)

Z x

a²+ x²dx= 1

2ln |a²+ x²| (10) Z x²

a² + x²dx= x − a tan⁻¹ x

a (11)

Z x³

a²+ x²dx= 1

2x²− 1

2a²ln |a² + x²| (12)

Z 1

ax² + bx + cdx= 2

√4ac − b² tan⁻¹ 2ax + b

√4ac − b² (13)

Z 1

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

b− alna+ x

b+ x, a6= b (14)

Z x

(x + a)²dx= a

a+ x + ln |a + x| (15)

Z x

ax²+ bx + cdx= 1

2aln |ax²+bx+c|− b a√

4ac − b² tan⁻¹ 2ax + b

√4ac − b² (16)

Integrals with Roots

Z √

x− a dx = 2

3(x − a)^3/2 (17)

Z 1

√x± a dx= 2√

x± a (18)

Z 1

√a− x dx= −2√

a− x (19)

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

(20)

Z √

ax+ b dx = 2b 3a + 2x

3

√

ax+ b (21)

Z

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

5a(ax + b)^5/2 (22)

Z x

√x± a dx= 2

3(x ∓ 2a)√

x± a (23)

Z r x

a− x dx= −p

x(a − x) − a tan⁻¹ px(a − x)

x− a (24)

Z r x

a+ x dx=p

x(a + x) − a ln√

x+√ x+ a

(25) Z

x√

ax+ b dx = 2

15a²(−2b²+ abx + 3a²x²)√

ax+ b (26)

Z

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

h(2ax + b)p

ax(ax + b) − b²ln  a√

x+p

a(ax + b)   i (27)

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)   (28)

Z √

x²± a² dx= 1 2x√

x²± a²± 1 2a²ln

 x+√

x² ± a² 

 (29)

Z √

a²− x² dx= 1 2x√

a²− x²+ 1

2a²tan⁻¹ x

√a²− x² (30) Z

x√

x²± a² dx= 1

3 x²± a²
3/2

(31)

Z 1

√x² ± a² dx = ln  x+√

x²± a² 

 (32)

Z 1

√a²− x² dx= sin⁻¹ x

a (33)

Z x

√x²± a² dx=√

x²± a² (34)

Z x

√a²− x² dx= −√

a²− x² (35)

Z x²

√x²± a² dx= 1 2x√

x²± a² ∓1 2a²ln

 x+√

x²± a² 

 (36)

Z √

ax²+ bx + c dx = b+ 2ax 4a

√ax²+ bx + c+4ac − b² 8a^3/2 ln

2ax+ b + 2p

a(ax²+ bx⁺c)   (37)

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)

Z 1

√ax² + bx + c dx= 1

√aln 

2ax+ b + 2p

a(ax²+ bx + c) 

 (39)

Z x

√ax²+ bx + c dx= 1 a

√ax²+ bx + c− b 2a^3/2 ln

2ax+ b + 2p

a(ax²+ bx + c)   (40)

Z dx

(a²+ x²)^3/2 = x a²√

a²+ x² (41)

Integrals with Logarithms

Z

ln ax dx = x ln ax − x (42)

Z

xln x dx = 1

2x²ln x − x²

4 (43)

Z

x²ln x dx = 1

3x³ln x −x³

9 (44)

Z

xⁿln x dx = xⁿ⁺¹

 ln x

n+ 1 − 1 (n + 1)²



, n6= −1 (45) Z ln ax

x dx= 1

2(ln ax)² (46)

Z ln x

x² dx= −1

x − ln x

x (47)

Z

ln(ax + b) dx =

 x+ b

a



ln(ax + b) − x, a 6= 0 (48) Z

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

a − 2x (49)

Z

ln(x²− a²) dx = x ln(x² − a²) + a lnx+ a

x− a − 2x (50) Z

ln ax²+ bx + c
dx = 1 a

√4ac − b²tan⁻¹ 2ax + b

√4ac − b²−2x+ b 2a + x



ln ax²+ bx + c
(51)

Z

xln(ax + b) dx = bx 2a − 1

4x²+ 1 2



x² − b² a²



ln(ax + b) (52)

Z

xln a² − b²x²

dx= −1

2x²+ 1 2



x²−a² b²



ln a²− b²x²

(53)

Z

(ln x)² dx= 2x − 2x ln x + x(ln x)² (54) Z

(ln x)³ dx= −6x + x(ln x)³− 3x(ln x)²+ 6x ln x (55) Z

x(ln x)² dx= x² 4 +1

2x²(ln x)²− 1

2x²ln x (56) Z

x²(ln x)² dx = 2x³ 27 +1

3x³(ln x)²− 2

9x³ln x (57)

Integrals with Exponentials

Z

e^ax dx= 1

ae^ax (58)

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 (59) Z

xe^x dx= (x − 1)e^x (60)

Z

xe^ax dx= x a − 1

a²



e^ax (61)

Z

x²e^x dx= x²− 2x + 2
e^x (62) Z

x²e^ax dx= x² a − 2x

a² + 2 a³



e^ax (63)

Z

x³e^x dx= x³− 3x²+ 6x − 6
e^x (64)

Z

xⁿe^ax dx= xⁿe^ax a −n

a Z

xⁿ⁻¹e^axdx (65)

Z

xⁿe^ax dx= (−1)ⁿ

aⁿ⁺¹ Γ[1 + n, −ax], where Γ(a, x) = Z ^∞

x

t^a−1e^−tdt (66) Z

e^ax2 dx= −i√ π 2√

aerf ix√ a

(67) Z

e^−ax2 dx=

√π

2√aerf x√ a

(68) Z

xe^−ax2 dx = − 1

2ae^−ax2 (69)

Z

x²e^−ax2 dx= 1 4

r π

a³erf(x√

a) − x

2ae^−ax2 (70)

Integrals with Trigonometric Functions

Z

sin ax dx = −1

acos ax (71)

Z

sin²ax dx= x

2 − sin 2ax

4a (72)

Z

sin³ax dx= −3 cos ax

4a +cos 3ax

12a (73)

Z

sinⁿax dx= −1

acos ax 2F₁ 1

2,1 − n 2 ,3

2,cos²ax



(74) Z

cos ax dx = 1

asin ax (75)

Z

cos²ax dx= x

2 + sin 2ax

4a (76)

Z

cos³axdx= 3 sin ax

4a +sin 3ax

12a (77)

Z

cos^paxdx= − 1

a(1 + p)cos^1+pax×2F₁ 1 + p 2 ,1

2,3 + p

2 ,cos²ax



(78)

Z

cos x sin x dx = 1

2sin²x+ c1 = −1

2cos²x+ c2 = −1

4cos 2x + c3 (79) Z

cos ax sin bx dx = cos[(a − b)x]

2(a − b) − cos[(a + b)x]

2(a + b) , a6= b (80) Z

sin²axcos bx dx = −sin[(2a − b)x]

4(2a − b) +sin bx

2b − sin[(2a + b)x]

4(2a + b) (81) Z

sin²xcos x dx = 1

3sin³x (82)

Z

cos²axsin bx dx = cos[(2a − b)x]

4(2a − b) −cos bx

2b − cos[(2a + b)x]

4(2a + b) (83) Z

cos²axsin ax dx = − 1

3acos³ax (84)

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) (85) Z

sin²axcos²ax dx= x

8 − sin 4ax

32a (86)

Z

tan ax dx = −1

aln cos ax (87)

Z

tan²ax dx= −x + 1

atan ax (88)

Z

tanⁿax dx= tanⁿ⁺¹ax

a(1 + n) ײF₁ n + 1

2 ,1,n+ 3

2 ,− tan²ax



(89)

Z

tan³axdx= 1

aln cos ax + 1

2asec²ax (90)

Z

sec x dx = ln | sec x + tan x| = 2 tanh⁻¹ tanx

2

 (91)

Z

sec²ax dx = 1

atan ax (92)

Z

sec³x dx= 1

2sec x tan x + 1

2ln | sec x + tan x| (93) Z

sec x tan x dx = sec x (94)

Z

sec²xtan x dx = 1

2sec²x (95)

Z

secⁿxtan x dx = 1

nsecⁿx, n6= 0 (96) Z

csc x dx = ln  tan

x 2  

 = ln | cscx− cot x| + C (97) Z

csc²ax dx = −1

acot ax (98)

Z

csc³x dx= −1

2cot x csc x + 1

2ln | csc x − cot x| (99) Z

cscⁿxcot x dx = −1

n cscⁿx, n6= 0 (100) Z

sec x csc x dx = ln | tan x| (101)

Products of Trigonometric Functions and Mono- mials

Z

xcos x dx = cos x + x sin x (102) Z

xcos ax dx = 1

a² cos ax + x

a sin ax (103)

Z

x²cos x dx = 2x cos x + x² − 2
sin x (104) Z

x²cos ax dx = 2x cos ax

a² + a²x²− 2

a³ sin ax (105)

Z

xⁿcos xdx = −1

2(i)ⁿ⁺¹[Γ(n + 1, −ix) + (−1)ⁿΓ(n + 1, ix)] (106)

Z

xⁿcos ax dx = 1

2(ia)¹⁻ⁿ[(−1)ⁿΓ(n + 1, −iax) − Γ(n + 1, ixa)] (107) Z

xsin x dx = −x cos x + sin x (108) Z

xsin ax dx = −xcos ax

a +sin ax

a² (109)

Z

x²sin x dx = 2 − x²
cos x + 2x sin x (110) Z

x²sin ax dx = 2 − a²x²

a³ cos ax + 2x sin ax

a² (111)

Z

xⁿsin x dx = −1

2(i)ⁿ[Γ(n + 1, −ix) − (−1)ⁿΓ(n + 1, −ix)] (112) Z

xcos²x dx= x² 4 + 1

8cos 2x + 1

4xsin 2x (113) Z

xsin²x dx= x² 4 − 1

8cos 2x − 1

4xsin 2x (114) Z

xtan²x dx= −x²

2 + ln cos x + x tan x (115) Z

xsec²x dx= ln cos x + x tan x (116)

Products of Trigonometric Functions and Ex- ponentials

Z

e^xsin x dx = 1

2e^x(sin x − cos x) (117)

Z

e^bxsin ax dx = 1

a² + b²e^bx(b sin ax − a cos ax) (118) Z

e^xcos x dx = 1

2e^x(sin x + cos x) (119) Z

e^bxcos ax dx = 1

a²+ b²e^bx(a sin ax + b cos ax) (120) Z

xe^xsin x dx = 1

2e^x(cos x − x cos x + x sin x) (121) Z

xe^xcos x dx = 1

2e^x(x cos x − sin x + x sin x) (122)

Integrals of Hyperbolic Functions

Z

cosh ax dx = 1

asinh ax (123)

Z

e^axcosh bx dx =





 e^ax

a²− b²[a cosh bx − b sinh bx] a 6= b e^2ax

4a + x

2 a= b

(124)

Z

sinh ax dx = 1

acosh ax (125)

Z

e^axsinh bx dx =





 e^ax

a²− b²[−b cosh bx + a sinh bx] a 6= b e^2ax

4a −x

2 a = b

(126)

Z

tanh ax dx = 1

aln cosh ax (127)

Z

e^axtanh bx dx =















e^(a+2b)x (a + 2b)²F₁h

1 + a

2b,1, 2 + a

2b,−e^2bxi

−1

ae^ax₂F₁h 1, a

2b,1 + a

2b,−e^2bxi

a6= b e^ax− 2 tan⁻¹[e^ax]

a a= b

(128)

Z

cos ax cosh bx dx = 1

a²+ b² [a sin ax cosh bx + b cos ax sinh bx] (129)

Z

cos ax sinh bx dx = 1

a²+ b² [b cos ax cosh bx + a sin ax sinh bx] (130)

Z

sin ax cosh bx dx = 1

a²+ b² [−a cos ax cosh bx + b sin ax sinh bx] (131)

Z

sin ax sinh bx dx = 1

a²+ b² [b cosh bx sin ax − a cos ax sinh bx] (132) Z

sinh ax cosh axdx = 1

4a[−2ax + sinh 2ax] (133) Z

sinh ax cosh bx dx = 1

b²− a² [b cosh bx sinh ax − a cosh ax sinh bx] (134)