Care must be taken in choosing the first
function and the second function Obviously, we must take that function as
the second function whose integral is well known to us �
[ ( )
( )]
( )
C
x
x
e
f x
f
x
dx
e f x dx
′
+
=
+
∫
∫
� Some special types of integrals
(i)
2
2
2
2
2
2
2
log
C
2
2
x
a
x
a
dx
x
a
x
x
a
−
=
−
−
+
−
+
∫
(ii)
2
2
2
2
2
2
2
log
C
2
2
x
a
x
a
dx
x
a
x
x
a
+
=
+
+
+
+
+
∫
(iii)
2
2
2
2
2
sin1
C
2
2
x
a
x
a
x
dx
a
x
a
−
−
=
−
+
+
∫
(iv) Integrals of the types 
2
2
or
dx
dx
ax
bx
c
ax
bx
c
+
+
+
+
∫
∫
can be
transformed into standard form by expressing
ax2 + bx + c = 
2
2
2
2
2
4
b
c
b
c
b
a x
x
a
x
a
a
a
a
a








+
+
=
+
+
−


















(v) Integrals of the types 
2
2
or
px
q dx
px
q dx
ax
bx
c
ax
bx
c
+
+
+
+
+
+
∫
∫
can be
358
MATHEMATICS
transformed into standard form by expressing
2
A
(
)
B
A (2
)
B
d
px
q
ax
bx
c
ax
b
dx
+
=
+
+
+
=
+
+
, where A and B are
determined by comparing coefficients on both sides � We have defined
( )
b
∫a f x dx
 as the area of the region bounded by the curve
y = f (x), a ≤ x ≤ b, the x-axis and the ordinates x = a and x = b