∫ Xn

x dx=

Xn

n +a

∫ Xn− 1

x dx

Integrals containingX=a+bxandY =α+βx, (b 6 = 0, β 6 = 0, ∆ =aβ−αb 6 = 0 )

∫ dx

XY =

1

∆ln|

Y

X|+C

∫ x dx

XY =

1

∆

[a

bln|X|−

α

βln|Y|

]

+C

∫ x (^2) dx
XY =
x
bβ=
a^2
b^2 ∆ln|X|+
α^2
β^2 ∆ln|Y|+C
111.
∫ dx
X^2 Y =
1
∆
( 1
X+
β
∆ln|
Y
X|
)
+C
112.
∫ x dx
X^2 Y =−
a
b∆X−
α
∆^2 ln|
Y
X|+C
113.
∫ x (^2) dx
X^2 Y =
a^2
b^2 ∆X+
1
∆^2
[
α^2
βln|Y|+
a(aβ− 2 αb)
b^2 ln|X|
]
+C
114.
∫ X
Ydx=
b
βx+
∆
β^2 ln|
Y
X|+C
115.
∫ √
XY dx=∆ + 2 4 bβbY
√
XY−∆
2
8 bβ
∫ dx
√
XY
116.
∫ dx
XnYm=
− 1
(m−1)∆Xn−^1 Ym−^1 +
(m+n−2)b
(m−1)∆
∫ dx
XnYm−^1 , m^6 = 1
117.
∫ dx
Y
√
X






√^2
−∆βtan
− 1 β
√
√ X
−∆β,+C^1 ∆β <^0
√^1
∆βln|
β
√
X−
√
∆β
β
√
X+√∆β
|+C 2 , ∆β > 0
118.
∫ dx
√
XY






√^2
−bβ
tan−^1
√
−βX
bY +C^1 , bβ <^0 , bY >^0
√^2
bβ
tanh−^1
√
βX
bY +C^2 , bβ >^0 , bY >^0
119.
∫ x dx
√
XY
=bβ^1
√
XY−(bα 2 +bβaβ)
∫ dx
√
XY
120.
∫ √Y
√
X
dx=^1 b
√
XY−∆ 2 b
∫ dx
√
XY
121.
∫ √X
Y dx=
2
β
√
X+∆β
∫ dx
Y
√
X
Appendix C