(39)
Z
1
√
ax
2
+ bx + c
dx =
1
√
a
ln
2ax + b +2
p
a(ax
2
+ bx + c)
(40)
Z
x
√
ax
2
+ bx + c
dx =
1
a
√
ax
2
+ bx + c-
b
2a
3/2
ln
2ax + b +2
p
a(ax
2
+ bx + c)
(41)
Z
dx
(a
2
+ x
2
)
3/2
=
x
a
2
√
a
2
+ x
2
Integrals with Logarithms
(42)
Z
ln ax dx = x ln ax - x
(43)
Z
x ln x dx =
1
2
x
2
ln x -
x
2
4
(44)
Z
x
2
ln x dx =
1
3
x
3
ln x -
x
3
9
(45)
Z
x
n
ln x dx = x
n+1
ln x
n +1
-
1
(n + 1)
2
, n 6= -1
(46)
Z
ln ax
x
dx =
1
2
(ln ax)
2
(47)
Z
ln x
x
2
dx = -
1
x
-
ln x
x
5
(48)
Z
ln(ax + b) dx =
x +
b
a
ln(ax + b) - x, a 6=0
(49)
Z
ln(x
2
+ a
2
) dx = x ln(x
2
+ a
2
)+2a tan
-1
x
a
- 2x
(50)
Z
ln(x
2
- a
2
) dx = x ln(x
2
- a
2
)+ a ln
x + a
x - a
- 2x
(51)
Z
ln
(
ax
2
+ bx + c
)
dx =
1
a
√
4ac - b
2
tan
-1
2ax + b
√
4ac - b
2
-2x+
b
2a
+ x
ln
(
ax
2
+ bx + c
)
(52)
Z
x ln(ax + b) dx =
bx
2a
-
1
4
x
2
+
1
2
x
2
-
b
2
a
2
ln(ax + b)
(53)
Z
x ln
(
a
2
- b
2
x
2
)
dx = -
1
2
x
2
+
1
2
x
2
-
a
2
b
2
ln
(
a
2
- b
2
x
2
)
(54)
Z
(ln x)
2
dx =2x - 2x ln x + x(ln x)
2
(55)
Z
(ln x)
3
dx = -6x + x(ln x)
3
- 3x(ln x)
2
+6x ln x
(56)
Z
x(ln x)
2
dx =
x
2
4
+
1
2
x
2
(ln x)
2
-
1
2
x
2
ln x
(57)
Z
x
2
(ln x)
2
dx =
2x
3
27
+
1
3
x
3
(ln x)
2
-
2
9
x
3
ln x
6