Page 1 of 8
INDEFINITE INTEGRATION FORMULAE:
x n1
1.  x dx 
 c, n  1 (NOTE:  dx  x  c )
n 1
1
2.  dx  log x  c
x
ax
x
3.  a dx 
c
log a
n
4.  e x dx  e x  c
5.  sin xdx   cos x  c
6.  cos xdx  sin x  c
7.  sec 2 x dx  tan x  c
8.  cos ec 2 x dx =  cotx +c
9.  sec x tan xdx  sec x  c
10.  cos ecx cot xdx   cos ecx  c
11.  tan xdx   log cos x  c or log sec x  c
12.  cot xdx  log sin x  c
 x 
13.  sec xdx  log sec x  tan x  c or log tan     c
 4 2
x
14.  cos ecxdx  log cos ecx  cot x  c or log tan  c
2
1
dx  sin 1 x + c
15. 
2
1 x
1
dx  cos 1 x + c
16.  
2
1 x
1
dx  tan 1 x  c
17. 
1  x2
1
dx  cot 1 x  c
18.  
2
1 x
1
dx  sec 1 x  c
19. 
2
x x 1
1
dx  cos ec 1 x  c
20.  
2
x x 1
Page 2 of 8
ALGEBRA OF INTEGRATION
  f  x   f  x dx   f  x  dx   f  x dx
2.  k . f  x dx  k  f  x  dx
1.
1
2
1
2
METHODS OF INTGRATION
1. Integration using formulae i.e. simple integration
2. Integration by substitution
(i)
Integrand of the form f  ax  b 
FORMULAE BASED ON f  ax  b 
1.
 ax  b 
ax

b
dx




a  n  1
2.
 ax  b dx 
n 1
n
1
log ax  b
a
 c , n  1
c
c ax b
k
a log c
eax b
4.  eax b dx 
c
a
cos  ax  b 
5.  sin  ax  b  dx  
c
a
sin  ax  b 
c
6.  cos  ax  b  dx 
a
tan  ax  b 
c
7.  sec2  ax  b  dx 
a
cot  ax  b 
+c
8.  cos ec 2  ax  b  dx = 
a
sec  ax  b 
c
9.  sec  ax  b  tan  ax  b  dx 
a
cos ec  ax  b 
c
10.  cos ec  ax  b  cot  ax  b  dx  
a
log cos ax  b
log sec ax  b
 c or
c
11.  tan  ax  b  dx  
a
a
log sin ax  b
c
12.  cot  ax  b  dx 
a
3.  c ax b dx 
   ax  b  
log tan  

2 
log sec  ax  b   tan  ax  b 
4
 c or
c
13.  sec  ax  b  dx 
a
a
Page 3 of 8
log cos ec  ax  b   cot  ax  b 
14.  cos ec  ax  b  dx 
15.

a
log tan
 c or
 ax  b 
2
a
sin 1  ax  b 
dx 
+c
2
a
1   ax  b 
c
1
cos 1  ax  b 
dx 
+c
16.  
2
a
1   ax  b 
1
17. 
1
1   ax  b 
18.  
19. 
2
dx 
1
1   ax  b 
2
tan 1  ax  b 
c
a
dx 
cot 1  ax  b 
c
a
1
 ax  b   ax  b 
20.  
2
1
dx 
1
 ax  b   ax  b 
2
1
(ii) Integration of the type
sec 1  ax  b 
c
a
dx 
cos ec 1  ax  b 
c
a
/
  f  x  . f  x  dx ; 
 g  f  x  . f  x  dx
n
f /  x
 f  x  
n
dx ;

f /  x
dx ;
f  x
/
METHOD: Put f(x) = t and f/(x) dx = dt and proceed.
NOTE:

f /  x
dx  log f  x   c
f  x
(iii) Integration of the type:  sin m  x  .cos n  x dx , where either ‘m’ or ‘n’ or both are
odd.
METHOD:
Case(i) If power of sine i.e. m is odd and power of cosine i.e. n is even then put
cos x  t and proceed.
Case(ii) If power of sine i.e. m is even and power of cosine i.e. n is odd then put
sin x  t and proceed.
Case(iii) If power of sine i.e. m is odd and power of cosine i.e. n is also odd then put
cos x  t or sin x  t and proceed.
Page 4 of 8
(iv). Integration which requires simplification by trigonometric functions:
Learn the following formulae:
1  cos 2 x
2
2sin A cos B  sin( A  B)  sin  A  B 
1  cos 2 x
2
1
sin 3 x  3sin x  sin 3 x 
4
1
cos3 x  3cos x  cos 3 x 
4
2cos A sin B  sin( A  B)  sin  A  B 
sin 2 x 
cos 2 x 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
NOTE: A student may require formulae of class XI, other then above; therefore he
is suggested to learn all trigonometric formulae studied in class XI.
(v). SOME SPECIAL INTEGRALS:
1
1
1
 bx  c 
 x
1. 
1. 
dx  sin 1 
dx  sin 1    c .
c.
2
2
b
a


a
a2  x2
a   bx  c 
2.
3. 
4.
1
 x
dx  cos1    c .
a
a x

2
1
1
x
dx  tan 1    c .
2
a x
a
a
2
a
5. 
2
2
1
1
x
dx  cot 1    c
2
x
a
a
1
 x
dx  sec1    c .
a
a
x x2  a2
1
6.
x
.
3. 
4.
1

a 2   bx  c 
1
a   bx  c 
2
2
2
1
 bx  c 
dx  cos 1 
c .
b
 a 
dx 
1
 bx  c 
tan 1 
c.
ab
 a 
dx 
1
 bx  c 
cot 1 
c
ab
 a 
1
 a  bx  c 
2
2
5.

1
 bx  c   bx  c 
2
 a2
dx 
1
 bx  c 
sec1 
c.
ab
 a 
dx 
1
 bx  c 
co sec1 
c
ab
 a 
6.
1
x a
2
2
dx 
1
 x
co sec1    c .
a
a

1
 bx  c  bx  c 
2
 a2
.
7.
7.

2.
1
x a
2
2
dx  log x  x 2  a 2  c

.
1
1
dx  log  bx  c  
b
 bx  c   a 2
2
 bx  c 
2
 a2  c
Page 5 of 8
8.

1
x a
2
2
dx  log x  x 2  a 2  c
8.

1
1
2
dx  log  bx  c    bx  c   a 2  c .
2
2
b
 bx  c   a
.
9. 
1
1
xa
dx 
log
c.
2
x a
2a
xa
10.
a
2
2
1
1
ax
dx 
log
c.
2
x
2a
ax
9. 
10.
1
 bx  c 
2
a
2
dx 
1
 a  bx  c 
2
2
 bx  c   a  c .
1
log
2ab
 bx  c   a
dx 
a   bx  c 
1
log
c .
2ab
a   bx  c 
3. INTEGRATION PARTIAL FRACTIONS:
FACTOR IN
CORRESPONDING PARTIAL FRACTION
THE
DENOMIANTO
R
( Linear factor)
A
ax  b
ax  b
Repeated linear
factor
A
B

2
ax  b  ax  b 2
(i)  ax  b 
(ii)  ax  b 
n
A3
An
A1
A2


 ... 
2
3
n
ax  b  ax  b   ax  b 
 ax  b 
Quadratic factor
ax 2  bx  c
Ax  B
ax  bx  c
2
Repeated
quadratic factor
(i)  ax 2  bx  c 
2
(ii)  ax 2  bx  c 
(i)
n
A1 x  B1
A2 x  B2

2
ax  bx  c  ax 2  bx  c 2
(ii)
A3 x  B3
An x  Bn
A1 x  B1
A2 x  B2


 ... 
2
3
n
2
ax  bx  c  ax 2  bx  c   ax 2  bx  c 
 ax 2  bx  c 
NOTE: Where A,B and Ai’s and Bi’s are real numbers and are to be calculated by
an appropriate method
Page 6 of 8
NOTE: If in an integration of the type
p  x
(i.e.) a rational expression
q  x
deg  p  x    deg  q  x   then we first divide p  x  by q  x  and write
p  x
as
q  x
p  x
remainder
 quotient 
and then proceed.
q  x
divisor
4. INTEGRATION BY PARTS:
Integration by parts is used in integrating functions of the type f  x  .g  x  as
follows.
I
st
d

function  II nd function  dx  I st function   II nd function  dx     I st function     II nd function  dx  dx
 dx

Where the Ist and IInd functions are decided in the order of ILATE;
I: Inverse trigonometric function
L: Logarithmic function
T: Trigonometric functions
A: Algebraic functions
E: Exponential Functions
There are three type of questions based on integration by parts:
TYPE1. Directly based on the formulae
 x sin xdx ;  logxdx ;   sin x  dx etc.
TYPE2: Integration of the type:  e sin bxdx ;  e
-1
Example:
2
ax
ax
TYPE3: Integration of the type:
x
/
x
 e  f  x   f  x   dx  e f  x   c
 e  kf  x   f  x   dx  e f  x   c
kx
/
kx
5. SOME MORE SPECIAL INTEGRALS

x x2  a2 a2
x
a  x dx 
 sin 1    c
2
2
a
2.

3.

x x2  a2 a2
x  a dx 
 log x  x 2  a 2  c
2
2
2
2
x x a
a2
x 2  a 2 dx 
 log x  x 2  a 2  c
2
2
1.
2
2
2
2
cos bxdx
Page 7 of 8
NOTE: SOME MORE SPECIAL INTEGRALS OF THE TYPE f(ax+b)
1.

2.

3.

2
2


1   bx  c   bx  c   a
a2
 bx  c  
a   bx  c  dx  
 sin 1 
  c
b
2
2
 a 


2
2


1   bx  c   bx  c   a a 2
2
2
2
2 
 log  bx  c    bx  c   a   c
 bx  c   a dx  
b
2
2


2
2


1   bx  c   bx  c   a a 2
2
2
2
2 
bx

c

a
dx


log
bx

c

bx

c

a



 


c
b
2
2


2
2
6. INTEGRATION OF THE TYPE:
x2  1
 x4  kx2  1 dx ;
x
4
1
dx
 kx 2  1
METHOD:
STEP1: Divide the Nr. and Dr. by x2. We get 1 
1
in the Nr.
x2
2
1

STEP2: Introduce  x   in the Dr.
x

1
STEP3: Put x   t , as per the situation and proceed.
x
-----------------------------------------------------------------------------------------------------------TYPES OF INTEGRATION OTHER THAN GIVEN IN THE N.C.E.R.T.
1. Integration of the type
1
1
1
 a  b sin 2 xdx ,  a  b cos 2 x dx ,  a sin 2 x  b cos2 xdx
1
  a sin x  b cos x  dx
2
METHOD:
Step1. Divide Nr. and Dr. by sin 2 x (or cos 2 x )
Step2. In the Dr. replace cos ec 2 x by 1  cot 2 x (or s ec 2 x by 1  tan 2 x ) and proceed.
2. Integration of the type
1
1
1
1
 a  b sin xdx ,  a  b cos xdx ,  a sin x  b cos x dx  a sin x  b cos x  c dx
Page 8 of 8
METHOD:
1  tan 2 x
2
2 dx
dx and cos x 
Step1. Replace sin x 
2 x
2 x
1  tan
1  tan
2
2
2 x
2 x
Step2. In the Nr. Replace 1  tan
.
 sec
2
2
Step3. Put tan x  t and proceed.
2
2 tan x
3. Integration of the type.
a sin x  b cos x
dx
TYPE:1. 
c sin x  d cos x
METHOD:
d
Put a sin x  b cos x    c sin x  d cos x     c sin x  d cos x 
dx
Where  and  are to be calculated by an appropriate method.
a sin x  b cos x  c
dx
d sin x  e cos x  f
METHOD:
d
Put a sin x  b cos x  c    d sin x  e cos x  f     d sin x  e cos x  f   
dx
Where  and  are to be calculated by an appropriate method.
TYPE:2. 
4. Integration of the type 
  x
dx , where P and Q are either linear polynomial and
P Q
quadratic polynomial alternately or simultaneously.
CASE(i) If P & Q both are linear then put Q  t 2 and proceed.
CASE(ii) If P is quadratic & Q is linear then put Q  t 2 and proceed.
1
CASE(iii) If P is linear and Q is quadratic function of x, we put P  .
t
1
CASE(iv) If P and Q both are pure quadratic of the form ax 2  b then put x  .
t