Handbook of mathematics for engineers and scienteists part 167 doc
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1130 INTEGRALS
2.
dx
(a + x)(b + x)
=
1
a – b
ln
b + x
a + x
, a ≠ b.Fora = b, see Integral 2 with n =–2 in
Paragraph T2.1.1-1.
3.
xdx
(a + x)(b + x)
=
1
a – b
a ln |a + x| – b ln |b + x|
.
4.
dx
(a + x)(b + x)
2
=
1
(b – a)(b + x)
+
1
(a – b)
2
ln
a + x
b + x
.
5.
xdx
(a + x)(b + x)
2
=
b
(a – b)(b + x)
–
a
(a – b)
2
ln
a + x
b + x
.
6.
x
2
dx
(a + x)(b + x)
2
=
b
2
(b – a)(b + x)
+
a
2
(a – b)
2
ln |a + x| +
b
2
– 2ab
(b – a)
2
ln |b + x|.
7.
dx
(a + x)
2
(b + x)
2
=–
1
(a – b)
2
1
a + x
+
1
b + x
+
2
(a – b)
3
ln
a + x
b + x
.
8.
xdx
(a + x)
2
(b + x)
2
=
1
(a – b)
2
a
a + x
+
b
b + x
+
a + b
(a – b)
3
ln
a + x
b + x
.
9.
x
2
dx
(a + x)
2
(b + x)
2
=–
1
(a – b)
2
a
2
a + x
+
b
2
b + x
+
2ab
(a – b)
3
ln
a + x
b + x
.
T2.1.1-3. Integrals involving a
2
+ x
2
.
1.
dx
a
2
+ x
2
=
1
a
arctan
x
a
.
2.
dx
(a
2
+ x
2
)
2
=
x
2a
2
(a
2
+ x
2
)
+
1
2a
3
arctan
x
a
.
3.
dx
(a
2
+ x
2
)
3
=
x
4a
2
(a
2
+ x
2
)
2
+
3x
8a
4
(a
2
+ x
2
)
+
3
8a
5
arctan
x
a
.
4.
dx
(a
2
+ x
2
)
n+1
=
x
2na
2
(a
2
+ x
2
)
n
+
2n – 1
2na
2
dx
(a
2
+ x
2
)
n
; n = 1, 2,
5.
xdx
a
2
+ x
2
=
1
2
ln(a
2
+ x
2
).
6.
xdx
(a
2
+ x
2
)
2
=–
1
2(a
2
+ x
2
)
.
7.
xdx
(a
2
+ x
2
)
3
=–
1
4(a
2
+ x
2
)
2
.
8.
xdx
(a
2
+ x
2
)
n+1
=–
1
2n(a
2
+ x
2
)
n
; n = 1, 2,
9.
x
2
dx
a
2
+ x
2
= x – a arctan
x
a
.
10.
x
2
dx
(a
2
+ x
2
)
2
=–
x
2(a
2
+ x
2
)
+
1
2a
arctan
x
a
.
T2.1. INDEFINITE INTEGRALS 1131
11.
x
2
dx
(a
2
+ x
2
)
3
=–
x
4(a
2
+ x
2
)
2
+
x
8a
2
(a
2
+ x
2
)
+
1
8a
3
arctan
x
a
.
12.
x
2
dx
(a
2
+ x
2
)
n+1
=–
x
2n(a
2
+ x
2
)
n
+
1
2n
dx
(a
2
+ x
2
)
n
; n = 1, 2,
13.
x
3
dx
a
2
+ x
2
=
x
2
2
–
a
2
2
ln(a
2
+ x
2
).
14.
x
3
dx
(a
2
+ x
2
)
2
=
a
2
2(a
2
+ x
2
)
+
1
2
ln(a
2
+ x
2
).
15.
x
3
dx
(a
2
+ x
2
)
n+1
=–
1
2(n – 1)(a
2
+ x
2
)
n–1
+
a
2
2n(a
2
+ x
2
)
n
; n = 2, 3,
16.
dx
x(a
2
+ x
2
)
=
1
2a
2
ln
x
2
a
2
+ x
2
.
17.
dx
x(a
2
+ x
2
)
2
=
1
2a
2
(a
2
+ x
2
)
+
1
2a
4
ln
x
2
a
2
+ x
2
.
18.
dx
x(a
2
+ x
2
)
3
=
1
4a
2
(a
2
+ x
2
)
2
+
1
2a
4
(a
2
+ x
2
)
+
1
2a
6
ln
x
2
a
2
+ x
2
.
19.
dx
x
2
(a
2
+ x
2
)
=–
1
a
2
x
–
1
a
3
arctan
x
a
.
20.
dx
x
2
(a
2
+ x
2
)
2
=–
1
a
4
x
–
x
2a
4
(a
2
+ x
2
)
–
3
2a
5
arctan
x
a
.
21.
dx
x
3
(a
2
+ x
2
)
2
=–
1
2a
4
x
2
–
1
2a
4
(a
2
+ x
2
)
–
1
a
6
ln
x
2
a
2
+ x
2
.
22.
dx
x
2
(a
2
+ x
2
)
3
=–
1
a
6
x
–
x
4a
4
(a
2
+ x
2
)
2
–
7x
8a
6
(a
2
+ x
2
)
–
15
8a
7
arctan
x
a
.
23.
dx
x
3
(a
2
+ x
2
)
3
=–
1
2a
6
x
2
–
1
a
6
(a
2
+ x
2
)
–
1
4a
4
(a
2
+ x
2
)
2
–
3
2a
8
ln
x
2
a
2
+ x
2
.
T2.1.1-4. Integrals involving a
2
– x
2
.
1.
dx
a
2
– x
2
=
1
2a
ln
a + x
a – x
.
2.
dx
(a
2
– x
2
)
2
=
x
2a
2
(a
2
– x
2
)
+
1
4a
3
ln
a + x
a – x
.
3.
dx
(a
2
– x
2
)
3
=
x
4a
2
(a
2
– x
2
)
2
+
3x
8a
4
(a
2
– x
2
)
+
3
16a
5
ln
a + x
a – x
.
4.
dx
(a
2
– x
2
)
n+1
=
x
2na
2
(a
2
– x
2
)
n
+
2n – 1
2na
2
dx
(a
2
– x
2
)
n
; n = 1, 2,
5.
xdx
a
2
– x
2
=–
1
2
ln |a
2
– x
2
|.
1132 INTEGRALS
6.
xdx
(a
2
– x
2
)
2
=
1
2(a
2
– x
2
)
.
7.
xdx
(a
2
– x
2
)
3
=
1
4(a
2
– x
2
)
2
.
8.
xdx
(a
2
– x
2
)
n+1
=
1
2n(a
2
– x
2
)
n
; n = 1, 2,
9.
x
2
dx
a
2
– x
2
=–x +
a
2
ln
a + x
a – x
.
10.
x
2
dx
(a
2
– x
2
)
2
=
x
2(a
2
– x
2
)
–
1
4a
ln
a + x
a – x
.
11.
x
2
dx
(a
2
– x
2
)
3
=
x
4(a
2
– x
2
)
2
–
x
8a
2
(a
2
– x
2
)
–
1
16a
3
ln
a + x
a – x
.
12.
x
2
dx
(a
2
– x
2
)
n+1
=
x
2n(a
2
– x
2
)
n
–
1
2n
dx
(a
2
– x
2
)
n
; n = 1, 2,
13.
x
3
dx
a
2
– x
2
=–
x
2
2
–
a
2
2
ln |a
2
– x
2
|.
14.
x
3
dx
(a
2
– x
2
)
2
=
a
2
2(a
2
– x
2
)
+
1
2
ln |a
2
– x
2
|.
15.
x
3
dx
(a
2
– x
2
)
n+1
=–
1
2(n – 1)(a
2
– x
2
)
n–1
+
a
2
2n(a
2
– x
2
)
n
; n = 2, 3,
16.
dx
x(a
2
– x
2
)
=
1
2a
2
ln
x
2
a
2
– x
2
.
17.
dx
x(a
2
– x
2
)
2
=
1
2a
2
(a
2
– x
2
)
+
1
2a
4
ln
x
2
a
2
– x
2
.
18.
dx
x(a
2
– x
2
)
3
=
1
4a
2
(a
2
– x
2
)
2
+
1
2a
4
(a
2
– x
2
)
+
1
2a
6
ln
x
2
a
2
– x
2
.
T2.1.1-5. Integrals involving a
3
+ x
3
.
1.
dx
a
3
+ x
3
=
1
6a
2
ln
(a + x)
2
a
2
– ax + x
2
+
1
a
2
√
3
arctan
2x – a
a
√
3
.
2.
dx
(a
3
+ x
3
)
2
=
x
3a
3
(a
3
+ x
3
)
+
2
3a
3
dx
a
3
+ x
3
.
3.
xdx
a
3
+ x
3
=
1
6a
ln
a
2
– ax + x
2
(a + x)
2
+
1
a
√
3
arctan
2x – a
a
√
3
.
4.
xdx
(a
3
+ x
3
)
2
=
x
2
3a
3
(a
3
+ x
3
)
+
1
3a
3
xdx
a
3
+ x
3
.
5.
x
2
dx
a
3
+ x
3
=
1
3
ln |a
3
+ x
3
|.
T2.1. INDEFINITE INTEGRALS 1133
6.
dx
x(a
3
+ x
3
)
=
1
3a
3
ln
x
3
a
3
+ x
3
.
7.
dx
x(a
3
+ x
3
)
2
=
1
3a
3
(a
3
+ x
3
)
+
1
3a
6
ln
x
3
a
3
+ x
3
.
8.
dx
x
2
(a
3
+ x
3
)
=–
1
a
3
x
–
1
a
3
xdx
a
3
+ x
3
.
9.
dx
x
2
(a
3
+ x
3
)
2
=–
1
a
6
x
–
x
2
3a
6
(a
3
+ x
3
)
–
4
3a
6
xdx
a
3
+ x
3
.
T2.1.1-6. Integrals involving a
3
– x
3
.
1.
dx
a
3
– x
3
=
1
6a
2
ln
a
2
+ ax + x
2
(a – x)
2
+
1
a
2
√
3
arctan
2x + a
a
√
3
.
2.
dx
(a
3
– x
3
)
2
=
x
3a
3
(a
3
– x
3
)
+
2
3a
3
dx
a
3
– x
3
.
3.
xdx
a
3
– x
3
=
1
6a
ln
a
2
+ ax + x
2
(a – x)
2
–
1
a
√
3
arctan
2x + a
a
√
3
.
4.
xdx
(a
3
– x
3
)
2
=
x
2
3a
3
(a
3
– x
3
)
+
1
3a
3
xdx
a
3
– x
3
.
5.
x
2
dx
a
3
– x
3
=–
1
3
ln |a
3
– x
3
|.
6.
dx
x(a
3
– x
3
)
=
1
3a
3
ln
x
3
a
3
– x
3
.
7.
dx
x(a
3
– x
3
)
2
=
1
3a
3
(a
3
– x
3
)
+
1
3a
6
ln
x
3
a
3
– x
3
.
8.
dx
x
2
(a
3
– x
3
)
=–
1
a
3
x
+
1
a
3
xdx
a
3
– x
3
.
9.
dx
x
2
(a
3
– x
3
)
2
=–
1
a
6
x
–
x
2
3a
6
(a
3
– x
3
)
+
4
3a
6
xdx
a
3
– x
3
.
T2.1.1-7. Integrals involving a
4
x
4
.
1.
dx
a
4
+ x
4
=
1
4a
3
√
2
ln
a
2
+ ax
√
2 + x
2
a
2
– ax
√
2 + x
2
+
1
2a
3
√
2
arctan
ax
√
2
a
2
– x
2
.
2.
xdx
a
4
+ x
4
=
1
2a
2
arctan
x
2
a
2
.
3.
x
2
dx
a
4
+ x
4
=–
1
4a
√
2
ln
a
2
+ ax
√
2 + x
2
a
2
– ax
√
2 + x
2
+
1
2a
√
2
arctan
ax
√
2
a
2
– x
2
.
1134 INTEGRALS
4.
dx
a
4
– x
4
=
1
4a
3
ln
a + x
a – x
+
1
2a
3
arctan
x
a
.
5.
xdx
a
4
– x
4
=
1
4a
2
ln
a
2
+ x
2
a
2
– x
2
.
6.
x
2
dx
a
4
– x
4
=
1
4a
ln
a + x
a – x
–
1
2a
arctan
x
a
.
T2.1.2. Integrals Involving Irrational Functions
T2.1.2-1. Integrals involving x
1/2
.
1.
x
1/2
dx
a
2
+ b
2
x
=
2
b
2
x
1/2
–
2a
b
3
arctan
bx
1/2
a
.
2.
x
3/2
dx
a
2
+ b
2
x
=
2x
3/2
3b
2
–
2a
2
x
1/2
b
4
+
2a
3
b
5
arctan
bx
1/2
a
.
3.
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.
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.
dx
(a
2
+ b
2
x)x
1/2
=
2
ab
arctan
bx
1/2
a
.
6.
dx
(a
2
+ b
2
x)x
3/2
=–
2
a
2
x
1/2
–
2b
a
3
arctan
bx
1/2
a
.
7.
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.
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.
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.
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.
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.
dx
(a
2
– b
2
x)x
1/2
=
1
ab
ln
a + bx
1/2
a – bx
1/2
.
13.
dx
(a
2
– b
2
x)x
3/2
=–
2
a
2
x
1/2
+
b
a
3
ln
a + bx
1/2
a – bx
1/2
.
14.
dx
(a
2
– b
2
x)
2
x
1/2
=
x
1/2
a
2
(a
2
– b
2
x)
+
1
2a
3
b
ln
a + bx
1/2
a – bx
1/2
.
T2.1. INDEFINITE INTEGRALS 1135
T2.1.2-2. Integrals involving (a + bx)
p/2
.
1.
(a + bx)
p/2
dx =
2
b(p + 2)
(a + bx)
(p+2)/2
.
2.
x(a + bx)
p/2
dx =
2
b
2
(a + bx)
(p+4)/2
p + 4
–
a(a + bx)
(p+2)/2
p + 2
.
3.
x
2
(a + bx)
p/2
dx =
2
b
3
(a + bx)
(p+6)/2
p + 6
–
2a(a + bx)
(p+4)/2
p + 4
+
a
2
(a + bx)
(p+2)/2
p + 2
.
T2.1.2-3. Integrals involving (x
2
+ a
2
)
1/2
.
1.
(x
2
+ a
2
)
1/2
dx =
1
2
x(a
2
+ x
2
)
1/2
+
a
2
2
ln
x +(x
2
+ a
2
)
1/2
.
2.
x(x
2
+ a
2
)
1/2
dx =
1
3
(a
2
+ x
2
)
3/2
.
3.
(x
2
+ a
2
)
3/2
dx =
1
4
x(a
2
+ x
2
)
3/2
+
3
8
a
2
x(a
2
+ x
2
)
1/2
+
3
8
a
4
ln
x +(x
2
+ a
2
)
1/2
.
4.
1
x
(x
2
+ a
2
)
1/2
dx =(a
2
+ x
2
)
1/2
– a ln
a +(x
2
+ a
2
)
1/2
x
.
5.
dx
√
x
2
+ a
2
=ln
x +(x
2
+ a
2
)
1/2
.
6.
xdx
√
x
2
+ a
2
=(x
2
+ a
2
)
1/2
.
7.
(x
2
+ a
2
)
–3/2
dx = a
–2
x(x
2
+ a
2
)
–1/2
.
T2.1.2-4. Integrals involving (x
2
– a
2
)
1/2
.
1.
(x
2
– a
2
)
1/2
dx =
1
2
x(x
2
– a
2
)
1/2
–
a
2
2
ln
x +(x
2
– a
2
)
1/2
.
2.
x(x
2
– a
2
)
1/2
dx =
1
3
(x
2
– a
2
)
3/2
.
3.
(x
2
– a
2
)
3/2
dx =
1
4
x(x
2
– a
2
)
3/2
–
3
8
a
2
x(x
2
– a
2
)
1/2
+
3
8
a
4
ln
x +(x
2
– a
2
)
1/2
.
4.
1
x
(x
2
– a
2
)
1/2
dx =(x
2
– a
2
)
1/2
– a arccos
a
x
.
5.
dx
√
x
2
– a
2
=ln
x +(x
2
– a
2
)
1/2
.
6.
xdx
√
x
2
– a
2
=(x
2
– a
2
)
1/2
.
7.
(x
2
– a
2
)
–3/2
dx =–a
–2
x(x
2
– a
2
)
–1/2
.
1136 INTEGRALS
T2.1.2-5. Integrals involving (a
2
– x
2
)
1/2
.
1.
(a
2
– x
2
)
1/2
dx =
1
2
x(a
2
– x
2
)
1/2
+
a
2
2
arcsin
x
a
.
2.
x(a
2
– x
2
)
1/2
dx =–
1
3
(a
2
– x
2
)
3/2
.
3.
(a
2
– x
2
)
3/2
dx =
1
4
x(a
2
– x
2
)
3/2
+
3
8
a
2
x(a
2
– x
2
)
1/2
+
3
8
a
4
arcsin
x
a
.
4.
1
x
(a
2
– x
2
)
1/2
dx =(a
2
– x
2
)
1/2
– a ln
a +(a
2
– x
2
)
1/2
x
.
5.
dx
√
a
2
– x
2
=arcsin
x
a
.
6.
xdx
√
a
2
– x
2
=–(a
2
– x
2
)
1/2
.
7.
(a
2
– x
2
)
–3/2
dx = a
–2
x(a
2
– x
2
)
–1/2
.
T2.1.2-6. Integrals involving arbitrary powers. Reduction formulas.
1.
dx
x(ax
n
+ b)
=
1
bn
ln
x
n
ax
n
+ b
.
2.
dx
x
√
x
n
+ a
2
=
2
an
ln
x
n/2
√
x
n
+ a
2
+ a
.
3.
dx
x
√
x
n
– a
2
=
2
an
arccos
a
x
n/2
.
4.
dx
x
√
ax
2n
+ bx
n
=–
2
√
ax
2n
+ bx
n
bnx
n
.
The parameters a, b, p, m, and n below in Integrals 5–8 can assume arbitrary values,
except for those at which denominators vanish in successive applications of a formula.
Notation: w = ax
n
+ b.
5.
x
m
(ax
n
+ b)
p
dx =
1
m + np + 1
x
m+1
w
p
+ npb
x
m
w
p–1
dx
.
6.
x
m
(ax
n
+ b)
p
dx =
1
bn(p + 1)
–x
m+1
w
p+1
+(m + n + np + 1)
x
m
w
p+1
dx
.
7.
x
m
(ax
n
+ b)
p
dx =
1
b(m + 1)
x
m+1
w
p+1
– a(m + n + np + 1)
x
m+n
w
p
dx
.
8.
x
m
(ax
n
+ b)
p
dx =
1
a(m + np + 1)
x
m–n+1
w
p+1
–b(m – n + 1)
x
m–n
w
p
dx
.
