Integrals containing terms of the forma+bxn∫ dx
a+bx^2 =
√^1
abtan−^1(√
b
ax)
+C, ab > 01
2√
−abln∣∣
∣∣a+√
−abx
a−√
−abx∣∣
∣∣+C, ab < 0∫ x dx
a+bx^2 =1
2 bln|x(^2) +a
b|+C
124.
∫ x (^2) dx
a+bx^2 =
x
b−
a
b
∫ dx
a+bx^2
125.
∫ dx
(a+bx^2 )^2 =
x
2 a(a+bx^2 )+
1
2 a
∫ dx
a+bx^2
126.
∫ dx
x(a+bx^2 )=
1
2 aln
∣∣
∣∣ x
2
a+bx^2
∣∣
∣∣+C
127.
∫ dx
x^2 (a+bx^2 )=−
1
ax−
b
a
∫ dx
a+bx^2
128.
∫ dx
(a+bx^2 )n+1=
1
2 na
x
(a+bx^2 )n+
2 n− 1
2 na
∫ dx
(a+bx^2 )n
129.
∫ dx
α^3 +β^3 x^3 =
1
6 α^2 β
[
2
√
3 tan−^1
( 2 βx−α
√
3 α
)
- ln
∣∣
∣∣ (α+βx)
2
α^2 −αβx+β^2 x^2
∣∣
∣∣
]
+C
- ∫ x dx
α^3 +β^3 x^3 =
1
6 αβ^2
[
2
√
3 tan−^1
( 2 βx−α
√
3 α
)
−ln
∣∣
∣∣ (α+βx)^2
α^2 −αβx+β^2 x^2
∣∣
∣∣
]
+C
IfX=a+bxn, then
∫
xm−^1 Xpdx= x
mXp
m+pn+
apn
m+pn
∫
xm−^1 Xp−^1 dx
∫
xm−^1 Xpdx=−x
mXp+1
an(p+ 1)+
m+pn+n
an(p+ 1)
∫
xm−^1 Xp+1dx
∫
xm−^1 XPdx=x
m−nXp+1
b(m+pn) −
(m−n)a
b(m+pn)
∫
xm−n−^1 Xpdx
∫
xm−^1 Xpdx=x
mXp+1
am −
(m+pn+n)b
am
∫
xm+n−^1 Xpdx
∫
xm−^1 Xpdx=x
m−nXp+1
bn(p+ 1) −
m−n
bn(p+ 1)
∫
xm−n−^1 Xp+1dx
∫
xm−^1 Xpdx=x
mXp
m −
bpn
m
∫
xm+n−^1 Xp−^1 dx
Appendix C