This document contains mathematical formulas and definitions related to calculus. It defines several functions of t including: 1. t2 2. Integrals of t from 0 to infinity for various functions of t. 3. Limits of these integrals as the bounds approach infinity.

1. No. )(tF )}({ tFL
1. 1 0,
1
>s
s
2. t
0,
1
2
>s
s
3. t 2
0,
2
3
>s
s
4. t n
n = 0,1,2,3,….
0,
!
1
>+
s
s
n
n
5.
at
e 0,
1
>
−
s
as
6. atsin
0,22
>
+
s
as
a
7. atcos
0,22
>
+
s
as
s
8. atsinh
as
as
a
>
−
,22
9. atcosh
as
as
s
>
−
,22
10. att cos
222
2
)( as
as
+
−
11.
a
att
2
sin
222
)( as
s
+

2. 1. 1)( =tF
∫
∞
−
==
`
0
)(1)}({ sfetFL st
∫
−
∞→
=
p
st
p
dteLim
0
p
st
p
e
s 0
1
lim 





−= −
∞→




+−= ∞−∞→ 0
11
lim
sesep
s
1
0 +=
s
1
=
)(sf=
2. ttF =)(
∫
∞
−
=
`
0
)}({ dttetFL st
( )st
p
p
ed
s
t −
∞→ ∫ −=
0
1
.lim
dtete
s
p
stst
p ∫
−−
∞→
−−=
0
lim
1
p
stst
p
e
s
te
s 0
1
lim
1




+−= −−
∞→
p
spsp
p
e
s
ee
s
pe
s 0
00 1
0
1
lim
1




+−



+−= −−
∞→
( ) 











+−+−=
ss
1
000
1






−−=
ss
1
0
1

3. 2
1
s
=
3. at
etF =)(
∫
∞
−
=
`
0
)}({ dtetetFL atst
dte
p
tas
p ∫
−−
∞→
=
0
)(
lim
[ ]ptas
p
e
as
0
)(
lim
1 −−
∞→−
=






−
−−
= −∞−∞→ 0)()(
11
lim
)(
1
asasp eeas
as −
=
1
4. attF sin)( =
dtetFL st
∫
∞
−
=
0
atsin)}({
∫ −= −
∞→
p
st
p
atd
a
eLim
0
)(cos
1
p
stst
p
eatd
a
eat
a
Lim
00
)(cos
1
.cos
1








+−= ∫
∞
−−
∞→
p
p
stst
p
dteat
a
s
eat
a
Lim
0
.cos.cos
1








−+−= ∫
∞
−−
∞→
p
stst
p
atd
a
e
a
s
eat
a
Lim
00
)(sin
1
..cos
1








−−= ∫
∞
−−
∞→
pp
ststst
p
edatate
a
s
eat
a
Lim
00
2
)(.sinsin(.cos
1








−−−= ∫
−−−
∞→
pp
ststst
p
seatate
a
s
eat
a
Lim
00
2
).sinsin(.cos
1








−−−−= ∫
−−−
∞→

4. pp
ststst
p
seat
a
s
ate
a
s
eat
a
Lim
00
2
2
2
).sinsin.cos
1








−−−= ∫
−−−
∞→
p
stst
p
eat
a
s
eat
asa
a
Lim
0
222
2
.sin.cos
1






−−
+
= −−
∞→






−−
+
= stst
ea
ats
ea
at
sa
a
.
sin.
.
cos
222
2
( ) 











−−−−
+
= 0
1
0022
2
asa
a






+
=
asa
a 1
22
2
22
sa
a
+
=
5. attF cos)( =
dtetFL st
∫
∞
−
=
0
atcos)}({
∫
−
∞→
=
p
st
p
atd
a
eLim
0
)(sin
1
p
stst
p
eatd
a
eat
a
Lim
00
)(sin
1
.sin
1








−= ∫
∞
−−
∞→
p
p
stst
p
dteat
a
s
eat
a
Lim
0
.sin.sin
1








+= ∫
∞
−−
∞→
p
stst
p
atd
a
e
a
s
eat
a
Lim
00
)cos(
1
..sin
1








−+= ∫
∞
−−
∞→
pp
ststst
p
edatate
a
s
eat
a
Lim
00
2
)(.cos)cos((.sin
1








−−−+= ∫
−−−
∞→
pp
ststst
p
dtseatate
a
s
eat
a
Lim
00
2
).cos)cos(.sin
1








−+−= ∫
−−−
∞→
pp
ststst
p
eat
a
s
ate
a
s
eat
a
Lim
00
2
2
2
).cos)cos(.sin
1








−−= ∫
−−−
∞→

5. p
stst
p
eat
a
s
eat
aas
a
Lim
0
222
2
.cos.sin
1






−
+
= −−
∞→






−
+
= stst
ea
ats
ea
at
as
a
.
cos.
.
sin
222
2
( ) 











−−−
+
= 222
2
000
a
s
as
a






+
= 222
2
a
s
as
a
22
as
a
+
=
F(t) = t2

6. 4.