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Moderately Extensive Table of Integral Formulas
What follows is a moderately extensive table of integrals formulas, collected from a number of sources and
arising in a variety of applications.1 The intent of compiling a table of formulas on-line in this fashion is to
facilitate student use across a variety of courses without having to rely on the vagaries of commercial book
publishing and frequent release of new editions. Considerable effort has been made (and will continue to be
made) to ensure that all of these formulas are correct, but that doesn’t guarantee there are no errors present.
If you do find an apparent error, an e-mail to [email protected] would certainly be appreciated.
(In all formulas below, symbols {a, b, c, …} from the beginning of the alphabet denote constants, symbols {f,
g, h, …} from the middle of the alphabet generally denote functions, and symbols {u, v, x, y, z} from the end
of the alphabet denote variables. Because this table is primarily intended for use by “novices”, we have
included the constant of integration explicitly using the upper case ‘C’ wherever relevant. In a very few
cases, formulas contain a lower case ‘c’ which is to be regarded as some other constant distinct from the
constant of integration.)
This table is divided into a number of sections, each containing integrals have some distinctive form.
Although the sections are not bookmarked in this MS WORD version of the table, we give the list of section
headings here in order to aid you in navigating through the table.
1) Elementary Forms
2) Forms Containing a  bx
(M001 – M010, page 1)
(M011 – M023, page 2)
3) Forms Containing a  bu
4) Forms containing a 2  u 2 , u 2  a 2
(M024 – M035, page 3)
(M036 – M039, page 3)
5) Forms containing
u 2  a2
6) Forms containing
a2  u 2
7) Forms containing
8) Forms containing
9)
10)
11)
12)
13)
14)
u
a
2
a
2
 u2
2
(M040 – M051, page 4)
(M052 – M061, page 4)


3
3
2
(M062 – M070, page 5)
2
(M071 – M078, page 6)
Forms containing au2  bu  c
Forms containing aun  b
Trigonometric Functions
Inverse Trigonometric Functions
Logarithmic Forms
Exponential Forms
(M079 – M080, page 6)
(M081 – M090, page 7)
(M091 – M140, page 7)
(M141 – M146, page 10)
(M147 – M151, page 10)
(M152 – M157, page 10)
1. Elementary Forms
M001.
 adu  au  C
M003.
 f  y  du  
M002.
 af u  du  a  f u  du
f y 
dy
du, where y  

y
du
1
At the time of writing, the goal was to give a somewhat more comprehensive listing of formulas that arise in a variety of
applications of integral calculus than is common in standard general textbooks, but stopping short of the very extensive
tabulations that are available in some references, including whole books that are compilations of formulas. The
numbering of formulas using the prefix ‘M’ for “moderate” from the title is prompted by the thought that some day it might
be useful to either produce a somewhat shorter table (or ‘S’) or perhaps even a much more extensive table (or ‘L’).
David W. Sabo (2000)
Table of Integrals (Moderate)
Page 1 of 10
M004.
  f  g  h   du   f du   g du   h du 
M005.
u
n
du 
M007.
e
u
du  eu  C
M009.
b
au
1 n 1
u  C,
n 1
bau
C
a ln b
du 
 n  1
 b  0, b  1
du
 ln u  C
u
M006.

M008.
e
M010.
 lnu du  u lnu  u  C
au
du 
1 au
e C
a
2. Forms Containing : a  bx
a  bu 
 n  1 b
n 1
M011.
 a  bu 
M012.
 a  bu  b ln a  bu  C
M013.
 u a  bu 
n
du
du 
 C,
 n  1
1
n
du 
1
a
n2
n 1
a  bu   2
a  bu   C,
b2  n  2
b  n  1
n  1,  2 
n 3
n2
n 1

a  bu 

1   a  bu 
2  a  bu 
 2a
a
 C
M014.  u  a  bu  du  3 
b  n3
n2
n 1 


n
2
du
M015.
 a  bu 
M017.
  a  bu 
M018.
 a  bu  b
2
u du
2


1
C
b  a  bu 
1
b2
u 2 du
1 
a2 
a

bu

2
a
ln
a

bu




 C
a  bu 
b3 
  a  bu 
M020.
 u  a  bu    a ln
M022.
 u  a  bu    au  a
Page 2 of 10
du
2
a 

ln a  bu  a  bu   C


M019.
2
u du
1
 a  bu  b a  bu   a ln a  bu   C
1 1
2

a  bu   2a  a  bu   a2 ln a  bu   C
3  
2


u 2 du
du
M016.
2

1
1
a  bu
C
u
b
2
ln
M021.
du
 u  a  bu 
2

1
1 a  bu
 ln
C
a  a  bu  a 2
u
a  bu
C
u
Table of Integrals (Moderate)
David W. Sabo (2000)
M023.
du
 u  a  bu 
2

2
a  2bu
2b a  bu
 3 ln
C
u
a u  a  bu  a
2
a  bu
3. Forms Containing:
M024.

3
2
a  bu du 
 a  bu  2  C
3b
M026.
u
2
M027.
u
M028.
u
M029.
u
M030.

M032.

a  bu du 
du
a  bu
du
a  bu

1

2
a


2 8a2  12abu  15b2u 2  a  bu 
105b3
ln
a
a  bu du 
n
M025.
a  bu  a
a  bu  a

a  bu
u 2 du
M034.

M035.
u

a  bu
du
a  bu
2
C
2
C
2
u n  a  bu 3 2  na u n 1 a  bu du 


b  2n  3  
2 a  bu
C
b

2 8a2  4abu  3b2u 2
15b

2
2
a  0
a  bu
du
du  2 a  bu  a 
u
u a  bu
du
15b
3
a  0
 C,
a  bu
 C,
a
tan1
3
 u a  bu du  
2  2a  3bu a  bu 

3
M031.

a  bu
a  bu b
du
du  
 
2
u
2 u a  bu
u
M033.

u du
M037.
a
a  bu

2  2a  bu 
3b 2
a  bu  C
a  bu  C
a  bu b
du


au
2a u a  bu
4. Forms containing: a 2  u 2 , u 2  a 2
du
1
u 
 tan1    C
 u2 a
a
M036.
a
2
M038.
u
2
M039.

du
1 u a

ln
 C,
 a2 2a u  a
du
a
2
u
2

2
David W. Sabo (2000)

u
2
 a2
2
du
1 a u

ln
 C,
 u 2 2a a  u
a

1
1
u
u 
tan1    2 2
C
3
2
a
2a
  2a a  u
Table of Integrals (Moderate)
Page 3 of 10
2
 u2

u 2  a2
5. Forms containing:
1
u u 2  a2  a2 ln u  u 2  a2   C

2 
M040.

M041.

M043.
1 a  u 2  a2


C
 u u 2  a2 a ln
u
u 2  a2 du 
du
u a
2
2
 ln u  u 2  a 2  C
M042.
u
du
u a
2
2

1
u 
sec 1    C
a
a
du

u 2  a2
a  u 2  a2
du  u 2  a 2  a ln
C
u
u
M045.



a
u 2  a2
u 
du  u 2  a2  a sec 1    C  u 2  a2  a tan1 
C
2
2
u
a
 u a 
M046.

M047.
u
M048.
u
M049.

M044.
M050.
M051.
u du
u a
2
 u 2  a2  C
u 2  a 2 du 
2
u

2
u 2  a2 du 
u 2 du
u 2  a2

du
2
u 2  a2

Page 4 of 10



3
u 2
u  a2
4
u 2
u  a2
2
C
2

3
2
a2
a4
u u 2  a2  ln u  u 2  a2  C
8
8
a2
ln u  u 2  a2  C
2
u 2  a2
C
a 2u
u 2  a2
u 2  a2
du


 ln u  u 2  a 2  C
u
u2
6. Forms containing:
M052.

1 2
u  a2
3
a 2  u 2 du 
a2  u 2
 u 
1
2
2
2
1
u a  u  a sin     C
2 
 a  
Table of Integrals (Moderate)
David W. Sabo (2000)
u
 sin1    C
a
a u
 
du
u
 cos1    C
a
 
M053.

M054.
u
M055.

a2  u 2
a  a2  u 2
du  a2  u 2  a ln
C
u
u
M056.

u du
M058.
2
2
2
 u a  u du  
2
1 a  a2  u 2
  ln
C
a
u
a2  u 2
du
  a2  u 2  C
a2  u 2
u 2 du
M059.

M060.
u

a2  u 2
du

M063.

u 2  a2

3
du
u
2
 a2

3
 u u
M066.
2
2
2
u u  a
M067.
M068.
2
 a2

3



a
2
2

u 2 du
u
2
a
2

3
3

u u 2  a2
David W. Sabo (2000)
3
2
3

2
a2
8
2
2
 u a  u du  

1 2
a  u2
3

3
2
C

 u 
2
2
2
1
u a  u  a sin     C

 a  
2

3
2
x
a2  u 2
a2  u 2
du  
 sin1    C
2
a
u
u
 

3a2u 2
3a 4
u  a2 
ln u  u 2  a 2   C
2
2


C

M064.

5
u 2
u  a2
6


2

2
1 2
u  a2
5
u a
2

2

u a
2
u


3
M061.

du 
2
du


2
2
u u  a

1
4
du 
2
 a2
2
u

M065.
u
du 
2

u 2
a  u2
4
a2  u 2
C
a 2u

a2  u 2
2
M057.
u
u 2
a2
a  u2 
sin1    C
a
2
2
 
7. Forms containing:
M062.
or
2
2
5

u du
u
2
 a2

3

2
1
u  a2
2
C
C
2
a2u 2
u  a2
24


3
2

a 4u 2
u  a2
16
a6
ln u  u 2  a2  C
16
 ln u  u 2  a 2  C
1
a2 u 2  a2

1 a  u 2  a2
ln
C
u
a3
Table of Integrals (Moderate)
Page 5 of 10
M069.
M070.


du


u u 2  a2


3

2
a
8. Forms containing:
M071.
 a
M072.

2
 u2
a
2
 u2
 u a
M075.
 u a
M077.
M078.
2



3
du 
2
du
M074.
M076.

2

3
 u2
2
a
2

3
3
u 2 du
a
2
 u2

3
2

3

u 2 a2  u 2

a u
2
2

 C

2

3
2

 u 
3a 2u
3a 4
a2  u 2 
sin1     C
 a 
2
2
 
C

1 2
a  u2
5
M073.

5
2

u du
a
2
 u2

3

2
1
a  u2
2
C
C

5
2

a 2u 2
a  u2
24


3
2

u
a 4u 2
a6
a  u2 
sin1    C
a
16
16
 
u
 sin1    C
a
a u
 
u
2
3
3

2

du


2

 u2
2
1
du   u a2  u 2
6

du
u a2  u 2
 u 2  a2
u


2
u

u  a2
du  
2

 u2
1
a4
2
1
2
2
u a  u
4 
2
3
u a
2
u

u 
sec 1    C
a
a
1

2
a
2
du
u 2 u 2  a2
1

3
1
a2 a2  u 2

2
2

1 a  a2  u 2
ln
C
u
a3
1  a2  u 2


u
a 4 

 C
a2  u 2 
u
9. Forms containing: au 2  bu  c
M079.
1
du
 au  bu  c 
2
b  4ac
2
ln
2au  b  b 2  4ac
2au  b  b  4ac
2
 2au  b
tan1 
2
4ac  b2
 4ac  b
2

Page 6 of 10
2
 C,
2au  b
b
2

  C,

 4ac  0
b
 C,
b
2
2
 4ac  0 
 4ac  0


Table of Integrals (Moderate)
David W. Sabo (2000)
M080.
 au
2
u du
1
b
du
 ln au 2  bu  c 
2

2a au  bu  c
 bu  c 2a
10. Forms containing: au n  b
du
 b 
tan1 
u  C,
 a 
ab


1
M081.
 a  bu
M082.
 a  bu
M083.
 a  bu
M084.
u 2 du
u a
du
 a  bu 2  b  b  a  bu 2
M085.

M086.
M087.
M088.
M089.
2

2

2

du
u du



du
 a  bu 
2
 a  bu 

n

n

n

u du
a  bu 
2
u 2 du
 a  bu 
2
du
ln
a  u ab
 ab  0 
 C,
a  u ab
1
ln a  bu 2  C
2b
2
du
2
1
2 ab
 ab  0 

u
2a a  bu

2

1
du

2a a  bu 2
u

2  n  1 a a  bu
2

n 1

1

2b  n  1 a  bu 2

n 1
u

2b  n  1 a  bu
1
u2
 u  a  bu   2a ln a  bu
2
2
2

n 1
2n  3
2  n  1 a

du
a  bu 
2
n 1
C

C
1
2b  n  1

du
 a  bu 
2
n 1
du
1
b
du
M090.
 u  a  bu    au  a  a  bu
2
2
2
11. Trigonometric Functions
M091.
 sin u du   cos u  C
M092.
 cos u du  sin u  C
M093.
 tan u du   ln cos u  C  ln sec u  C
M094.
 cot u du  ln sin u  C   ln csc u  C
M095.
 sec u du  ln sec u  tan u  C
M096.
 csc u du  ln csc u  cot u  C
M098.
 sin
2
1
1
1
1
u du   cos u sin u  u  C  u  sin2u  C
2
2
2
4
David W. Sabo (2000)
Table of Integrals (Moderate)
Page 7 of 10


1
u du   cos u sin2 u  2  C
3
M099.
 sin
M101.
n
 sin u du  
M102.
 sin
M103.
 cos
M104.
 cos
M106.
3
du
2
u
M100.
1
1
1
1
sin u cos u  u  C  u  sin2u  C
2
2
2
4
3
u du 
1
sin u cos2 u  2  C
3
 cos
n
u du 
1
n 1
cosn 1 u sin u 
cosn  2 u du
n
n 
M107.
 tan
2
u du  tan u  u  C
M109.
 tan
M111.
n
 cot u du  
M112.
 sec
2
u du  tan u  C
M113.
 sec
n
u du 
M114.
 csc
2
u du   cot u  C
M115.
 csc
n
u du  
M116.
 sin  a  bu  du   b cos  a  bu   C
M118.
 cos
M120.
 1  cos u  tan  2   C
u du 


tann 1 u
  tann  2 u du
n 1
cot n 1 u
  cot n  2 u du,
n 1
Page 8 of 10
u
du
3u sin2u sin 4u


C
8
4
32
u du 
3u sin2u sin4u


C
8
4
32
1
tan2 u  ln cos u  C
2
M105.
 cos
M108.
 tan
3
u du 
M110.
 cot
2
u du   cot u  u  C
4
 n  1
1
n2
secn 2 u tan u 
secn 2 u du  C
n 1
n 1 
1
n2
csc n 2 u cot u 
csc n 2 u du  C
n 1
n 1 
1
2
u du 
  cot u  C
u du 
du
4
sinn 1 u cos u n  1

sinn  2 u du
n
n 
2
n
 sin
  sec 2u du  tan u  C
u 
1
M117.
 cos  a  bu  du  b sin  a  bu   C
M119.
 1 sin u 
M121.
 1  cos u   cot  2   C
Table of Integrals (Moderate)
du
du

tan 
4
u
C
2 
u 
David W. Sabo (2000)
1
M122.
 sin u cos u du  2 sin
M124.
 sin mu sin nu du 
M125.
 cos mu cos nu du 
M126.
 sin mu cos nu du  
M127.
 u sin  au  du  a
M128.
2
 u sin u du 
M130.
2
 u cos  au  du 
M131.
u
M132.
2
 u cos u du 
M133.
2
 u sin  au  du 
M134.
u
M135.
2
2
 u sin u du 
M136.
2
2
 u cos u du 
M137.
 sin u cos
M139.
 cos
M140.
 cos
1
n
n
2
2
u C
M123.
sin  m  n  u
2m  n

sin  m  n  u
2m  n
sin  m  n  u
2m  n

cos  m  n  u
2m  n
sin  au  
 C,
sin  m  n  u
2m  n

2m  n 
 C,
2
m
 n2
2

 n2
m
2

 n2

u
cos  au   C
a
u 2 u sin2u cos2u


C
4
4
8
cos  au  du 
m
 C,
cos  m  n  u
du
 sin u cos u  ln tan u  C
M129.
1
 u cos  au  du  a
2
cos  au  
u
sin  au   C
a
2u
a 2u 2  2
cos
au

sin  au   C


a2
a3
1 n
n
u sin  au    u n 1 sin  au  du
a
a
u 2 u sin2u cos2u


C
4
4
8
2u
a 2u 2  2
sin
au

cos  au   C


a2
a3
1
n
sin  au  du   u n cos  au    u n 1 cos au  du
a
a
m
m
n
u3  u2 1 
u cos2u

  sin2u 
C
6  4 8
4
u3  u2 1 
u cos2u

  sin2u 
C
6  4 8
4
u du  
u sinn u du 
1
cosn 1 u  C
n 1
 cos u sin
n
u du 
1
m 1
cosm 1 u sinn 1 u 
cosm 2 u sinn u du,
mn
mn 
u sinn u du  
David W. Sabo (2000)
M138.
1
n 1
cosm 1 u sinn 1 u 
cosm u sinn 2 u du,
mn
mn 
Table of Integrals (Moderate)
1
sinn 1 u  C
n 1
 m  n 
 m  n 
Page 9 of 10
12. Inverse Trigonometric Functions
M141.
 sin
M143.
 tan
M145.
 sec
1
u du  u sin1 u  1  u 2  C
M142.
 cos
1
u du  u tan1 u  ln 1  u 2  C
2
M144.
 cot
M146.
csc
M148.
 u ln u du 
1
1
u du  u sec 1 u  ln u  u 2  1  C
1
1
u du  u cos1 u  1  u 2  C
1
u du  u cot 1 u  ln 1  u 2  C
2
1
u du  u csc 1 u  ln u  u 2  1  C
13. Logarithmic Forms
M147.
 ln u du  u ln u  u  C
M149.
u
M150.
u
u
 u ln u  du  n  1ln u    n  1
2
ln u du 
u2
u2
ln u 
C
2
4
u3
u3
ln u 
C
3
9
n 1
n 1
n
2
 C,
 n  1
ln u 
n 1
1
M151.  
ln u   C
du 
u
n 1
n
14. Exponential Forms
M152.
e
M154.
u e
M155.
n au
 u e du 
u
du  e u  C
au
du 
e
au
du 
1 au
e C
a
au  1 au
e C
a2
u neau n n 1 au
  u e du  C  eau
a
a
M156.
au
 e sin  bu  du 
M157.
au
 e cos  bu  du 
Page 10 of 10
M153.
n
n ! u n k
  1 n  k !a
k
k 0
k 1
C
eau a sin  bu   b cos  bu  
C
a2  b2
eau a cos  bu   b sin  bu  
C
a2  b2
Table of Integrals (Moderate)
David W. Sabo (2000)