Begin2.DVI

109. 
     

∫∞

0

e−axsinxbxdx= tan−^1 ab

110. 
     

∫∞

0

e−ax−e−bx

x dx= ln

b

a

111. 
     

∫∞

0

e−a^2 x^2 cosbx dx=

√π

2 ae

−b^2 / 4 a^2

112. 
     

∫∞

0

e−(ax^2 +b/x^2 )dx=^12

√

π

ae

− 2 √ab

113. 
     

∫∞

0

x^2 ne−βx

2

dx=(2n−1)(2 2 nn+1−β3)n···^5 ·^3 ·^1

√π

β

114. 
     

∫∞

0

e−k

(x 2

a^2 +b

2

x^2

)

dx=

√π

2

√a

k

e−^2 kb/a
115.

∫∞

0

sinrx dx

x(x^4 + 2a^2 x^2 cos 2β+a^4 )=

π

2 a^4

[

1 −sin(arsin 2sinββ+ 2β)e−βrcosβ

]

116. 
     

∫∞

0

cosrx dx

x^4 + 2a^2 x^2 cos 2β+a^4 =

π

2 a^3

sin(β+arsinβ)

sin 2β e

−arcosβ

117. 
     

∫∞

0

sinrx dx

x(x^6 +a^6 )=

π

6 a^6

[

3 −e−ar− 2 e−ar/^2 cosar

√

3

2

]

118. 
     

∫∞

0

cosrx dx

x^6 +a^6 =

π

6 a^5

[

e−ar− 2 e−ar/^2 cos(ar

√

3

2 +

2 π

3 )

]

119. 
     

∫∞

0

sinπx dx

x(1−x^2 )=π

120. 
     

∫∞

0

e−qx−e−px

x cosbx dx=

1

2 ln

∣∣

∣∣p

(^2) +b 2
q^2 +b^2
∣∣
∣∣
121.
∫∞
0
e−qx−e−px
x sinbx dx= tan
− 1 p
b−tan
− 1 q
b
122.
∫∞
0
e−axsinpx−xsinqxdx= tan−^1 pa−tan−^1 qb
123.
∫∞
0
e−axcospx−xcosqxdx=^12 ln
∣∣
∣∣a
(^2) +a 2
a^2 +p^2
∣∣
∣∣
124.
∫∞
0
xe−x^2 sinax dx=a
√π
4 e
−a^2 / 4
125.
∫∞
0
x^2 e−x^2 cosax dx=
√π
4
(
1 −a
2
2
)
e−a^2 /^4
126.
∫∞
0
x^3 e−x^2 sinax dx=
√π
8
(
3 a−a
3
2
)
e−a^2 /^4
Appendix C