Table 7 1
Symbols/Terms/Phrases
Meaning
( )
∫f x dx
Integral of f with respect to x
f (x) in 
( )
∫f x dx
Integrand
x in  
( )
∫f x dx
Variable of integration
Integrate
Find the  integral
An integral of f
A function F such that
F′(x) = f (x)
Integration
The process of finding the integral
Constant of Integration
Any real number C, considered as
constant function
290
MATHEMATICS
We already know the formulae for the derivatives of many important functions From these formulae, we can write down immediately the corresponding formulae
(referred to as standard formulae) for the integrals of these functions, as listed below
which will be used to find integrals of other functions Derivatives
Integrals (Anti derivatives)
(i)
1
1
n
n
d
x
x
dx
n
+

 =


+


 ;
1
C
1
n
n
x
x dx
n
+
=
+
+
∫
, n ≠ –1
Particularly, we note that
( )
1
d
dxx
=  ;
      
C
dx
=x
+
∫
(ii)
(
sin)
cos
d
x
x
dx
=
 ;
cos
sin
C
x dx
x
=
+
∫
(iii)
(
– cos)
sin
d
x
x
dx
=
 ;
sin
cos
C
x dx
–
x
=
+
∫
(iv)
(
)
2
tan
sec
d
x
x
dx
=
 ;
sec2
tan
C
x dx
x
=
+
∫
(v)
(
)
2
– cot
cosec
d
x
x
dx
=
 ;
cosec2
cot
C
x dx
–
x
=
+
∫
(vi)
(
sec)
sec
tan
d
x
x
x
dx
=
 ;
sec
tan
sec
C
x
x dx
x
=
+
∫
(vii)
(
– cosec)
cosec
cot
d
x
x
x
dx
=
 ;
cosec
cot
– cosec
C
x
x dx
x
=
+
∫
(viii)
(
)
– 1
2
1
sin
1
d
x
dx
– x
=
 ;
– 1
2
sin
C
1
dx
x
– x
=
+
∫
(ix)
(
)
– 1
2
1
– cos
1
d
x
dx
– x
=
 ;
– 1
2
cos
C
1
dx
–
x
– x
=
+
∫
(x)
(
)
– 1
2
1
tan
1
d
x
dx
x
=
+
 ;
– 1
2
tan
C
1
dx
x
x
=
+
+
∫
(xi)
(
)
– 1
2
1
– cot
1
d
x
dx
x
=
+
 ;
– 1
2
cot
C
1
dx
–
x
x
=
+
∫+
INTEGRALS         291
(xii)
(
)
– 1
12
sec
1
d
x
dx
x
x –
=
 ;
– 1
2
sec
C
1
dx
x
x
x –
=
+
∫
(xiii)
(
)
– 1
12
– cosec
1
d
x
dx
x
x –
=
 ;
– 1
2
cosec
C
1
dx
–
x
x
x –
=
+
∫
(xiv)
(
x)
x
d
e
e
dx
=
 ;
C
x
x
e dx
=e
+
∫
(xv)
(
)
1
log|
|
d
x
dx
=x
;
1
log|
| C
dx
x
x
=
+
∫
(xvi)
x
x
d
a
a
dx
log a

 =




 ;
C
x
x
a
a dx
log a
=
+
∫
�Note  In practice, we normally do not mention the interval over which the various
functions are defined