In mathematics, a quadratic integral is an integral of the form
∫
d
x
a
+
b
x
+
c
x
2
.
It can be evaluated by completing the square in the denominator.
∫
d
x
a
+
b
x
+
c
x
2
=
1
c
∫
d
x
(
x
+
b
2
c
)
2
+
(
a
c
−
b
2
4
c
2
)
.
Assume that the discriminant q = b2 − 4ac is positive. In that case, define u and A by
u
=
x
+
b
2
c
,
and
−
A
2
=
a
c
−
b
2
4
c
2
=
1
4
c
2
(
4
a
c
−
b
2
)
.
The quadratic integral can now be written as
∫
d
x
a
+
b
x
+
c
x
2
=
1
c
∫
d
u
u
2
−
A
2
=
1
c
∫
d
u
(
u
+
A
)
(
u
−
A
)
.
The partial fraction decomposition
1
(
u
+
A
)
(
u
−
A
)
=
1
2
A
(
1
u
−
A
−
1
u
+
A
)
allows us to evaluate the integral:
1
c
∫
d
u
(
u
+
A
)
(
u
−
A
)
=
1
2
A
c
ln
(
u
−
A
u
+
A
)
+
constant
.
The final result for the original integral, under the assumption that q > 0, is
∫
d
x
a
+
b
x
+
c
x
2
=
1
q
ln
(
2
c
x
+
b
−
q
2
c
x
+
b
+
q
)
+
constant, where
q
=
b
2
−
4
a
c
.
This (hastily written) section may need attention.
In case the discriminant q = b2 − 4ac is negative, the second term in the denominator in
∫
d
x
a
+
b
x
+
c
x
2
=
1
c
∫
d
x
(
x
+
b
2
c
)
2
+
(
a
c
−
b
2
4
c
2
)
.
is positive. Then the integral becomes
1
c
∫
d
u
u
2
+
A
2
=
1
c
A
∫
d
u
/
A
(
u
/
A
)
2
+
1
=
1
c
A
∫
d
w
w
2
+
1
=
1
c
A
arctan
(
w
)
+
c
o
n
s
t
a
n
t
=
1
c
A
arctan
(
u
A
)
+
constant
=
1
c
a
c
−
b
2
4
c
2
arctan
(
x
+
b
2
c
a
c
−
b
2
4
c
2
)
+
constant
=
2
4
a
c
−
b
2
arctan
(
2
c
x
+
b
4
a
c
−
b
2
)
+
constant
.
