The method in which we change the variable to some
other variable is called the method of substitution When the integrand involves
some trigonometric functions, we use some well known identities to find the
integrals Using substitution technique, we obtain the following standard
integrals (i)
tan
log sec
C
x dx
x
=
+
∫
(ii)
cot
log sin
C
x dx
x
=
+
∫
(iii)
sec
log sec
tan
C
x dx
x
x
=
+
+
∫
(iv)
cosec
log cosec
cot
C
x dx
x
x
=
−
+
∫
� Integrals of some special functions
(i)
2
2
1 log
C
2
dx
x
a
a
x
a
x
a
−
=
+
+
−
∫
(ii)
2
2
1 log
C
2
dx
a
x
a
a
x
a
x
+
=
+
−
−
∫
(iii)
1
2
2
1 tan
C
dx
x
a
a
x
a
−
=
+
+
∫
INTEGRALS         357
(iv)
2
2
2
2
log
C
dx
x
x
a
x
a
=
+
−
+
−
∫
(v)
1
2
2
sin
C
dx
ax
a
x
−
=
+
−
∫
(vi)
2
2
2
2
log |
|
C
dx
x
x
a
x
a
=
+
+
+
+
∫
� Integration by parts
For given functions f1 and  f2, we have
1
2
1
2
1
2
( )
( )
( )
( )
( )
( )
d
f
x
f
x dx
f
x
f
x dx
f x
f
x dx dx
dx


⋅
=
−
⋅




∫
∫
∫
∫
, i