
1440 INDEFINITE AND DEFINITE INTEGRALS
3.
Z
x
1/2
dx
(a
2
+ b
2
x)
2
= −
x
1/2
b
2
(a
2
+ b
2
x)
+
1
ab
3
arctan
bx
1/2
a
.
4.
Z
x
3/2
dx
(a
2
+ b
2
x)
2
=
2x
3/2
b
2
(a
2
+ b
2
x)
+
3a
2
x
1/2
b
4
(a
2
+ b
2
x)
−
3a
b
5
arctan
bx
1/2
a
.
5.
Z
dx
(a
2
+ b
2
x)x
1/2
=
2
ab
arctan
bx
1/2
a
.
6.
Z
dx
(a
2
+ b
2
x)x
3/2
= −
2
a
2
x
1/2
−
2b
a
3
arctan
bx
1/2
a
.
7.
Z
dx
(a
2
+ b
2
x)
2
x
1/2
=
x
1/2
a
2
(a
2
+ b
2
x)
+
1
a
3
b
arctan
bx
1/2
a
.
8.
Z
x
1/2
dx
a
2
− b
2
x
= −
2
b
2
x
1/2
+
2a
b
3
ln
a + bx
1/2
a −bx
1/2
.
9.
Z
x
3/2
dx
a
2
− b
2
x
= −
2x
3/2
3b
2
−
2a
2
x
1/2
b
4
+
a
3
b
5
ln
a + bx
1/2
a −bx
1/2
.
10.
Z
x
1/2
dx
(a
2
−b
2
x)
2
=
x
1/2
b
2
(a
2
− b
2
x)
−
1
2ab
3
ln
a + bx
1/2
a −bx
1/2
.
11.
Z
x
3/2
dx
(a
2
−b
2
x)
2
=
3a
2
x
1/2
− 2b
2
x
3/2
b
4
(a
2
− b
2
x)
−
3a
2b
5
ln
a + bx
1/2
a − bx
1/2
.
12.
Z
dx
(a
2
−b
2
x)x
1/2
=
1
ab
ln
a + bx
1/2
a − bx
1/2
.
13.
Z
dx
(a
2
−b
2
x)x
3/2
= −
2
a
2
x
1