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 e