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 However, in any specific problem one has to keep it in mind 7 2