
Chapter 2 7
Indefinite and Definite Integrals
27.1 Indefinite Integrals
⋆
Throughout Section 27.1, the integration constant
C
is omitted for brevity.
27.1.1 In tegrals Involving Rational Functions
◮ Integrals involving a + bx.
1.
Z
dx
a + bx
=
1
b
ln |a + bx|.
2.
Z
(a + bx)
n
dx =
(a + bx)
n+1
b(n + 1)
, n 6= −1.
3.
Z
x dx
a + bx
=
1
b
2
a + bx − a ln |a + bx|
.
4.
Z
x
2
dx
a + bx
=
1
b
3
h
1
2
(a + bx)
2
− 2a(a + bx) + a
2
ln |a + bx|
i
.
5.
Z
dx
x(a + bx)
= −
1
a
ln
a + bx
x
.
6.
Z
dx
x
2
(a + bx)
= −
1
ax
+
b
a
2
ln
a + bx
x
.
7.
Z
x dx
(a + bx)
2
=
1
b
2
ln |a + bx| +
a
a + bx
.
8.
Z
x
2
dx
(a + bx)
2
=
1
b
3
a + bx −2a ln |a + bx|−
a
2
a + bx
.
9.
Z
dx
x(a + bx)
2
=
1
a(a + bx)
−
1
a
2
ln
a + bx
x
.
10.
Z
x dx
(a + bx)
3
=
1
b
2
h
−
1
a + bx
+
a
2(a + bx)