<s>
In	O
calculus	O
,	O
an	O
antiderivative	B-Algorithm
,	O
inverse	O
derivative	B-Algorithm
,	O
primitive	B-Algorithm
function	I-Algorithm
,	O
primitive	B-Algorithm
integral	I-Algorithm
or	O
indefinite	B-Algorithm
integral	I-Algorithm
of	O
a	O
function	O
is	O
a	O
differentiable	O
function	O
whose	O
derivative	B-Algorithm
is	O
equal	O
to	O
the	O
original	O
function	O
.	O
</s>
<s>
The	O
process	O
of	O
solving	O
for	O
antiderivatives	B-Algorithm
is	O
called	O
antidifferentiation	B-Algorithm
(	O
or	O
indefinite	B-Algorithm
integration	I-Algorithm
)	O
,	O
and	O
its	O
opposite	O
operation	O
is	O
called	O
differentiation	O
,	O
which	O
is	O
the	O
process	O
of	O
finding	O
a	O
derivative	B-Algorithm
.	O
</s>
<s>
Antiderivatives	B-Algorithm
are	O
often	O
denoted	O
by	O
capital	O
Roman	O
letters	O
such	O
as	O
and	O
.	O
</s>
<s>
Antiderivatives	B-Algorithm
are	O
related	O
to	O
definite	B-Algorithm
integrals	I-Algorithm
through	O
the	O
second	O
fundamental	O
theorem	O
of	O
calculus	O
:	O
the	O
definite	B-Algorithm
integral	I-Algorithm
of	O
a	O
function	O
over	O
a	O
closed	O
interval	O
where	O
the	O
function	O
is	O
Riemann	O
integrable	O
is	O
equal	O
to	O
the	O
difference	O
between	O
the	O
values	O
of	O
an	O
antiderivative	B-Algorithm
evaluated	O
at	O
the	O
endpoints	O
of	O
the	O
interval	O
.	O
</s>
<s>
In	O
physics	O
,	O
antiderivatives	B-Algorithm
arise	O
in	O
the	O
context	O
of	O
rectilinear	O
motion	O
(	O
e.g.	O
,	O
in	O
explaining	O
the	O
relationship	O
between	O
position	O
,	O
velocity	O
and	O
acceleration	O
)	O
.	O
</s>
<s>
The	O
discrete	O
equivalent	O
of	O
the	O
notion	O
of	O
antiderivative	B-Algorithm
is	O
antidifference	B-Algorithm
.	O
</s>
<s>
The	O
function	O
is	O
an	O
antiderivative	B-Algorithm
of	O
,	O
since	O
the	O
derivative	B-Algorithm
of	O
is	O
,	O
and	O
since	O
the	O
derivative	B-Algorithm
of	O
a	O
constant	O
is	O
zero	O
,	O
will	O
have	O
an	O
infinite	O
number	O
of	O
antiderivatives	B-Algorithm
,	O
such	O
as	O
,	O
etc	O
.	O
</s>
<s>
Thus	O
,	O
all	O
the	O
antiderivatives	B-Algorithm
of	O
can	O
be	O
obtained	O
by	O
changing	O
the	O
value	O
of	O
in	O
,	O
where	O
is	O
an	O
arbitrary	O
constant	O
known	O
as	O
the	O
constant	O
of	O
integration	O
.	O
</s>
<s>
Essentially	O
,	O
the	O
graphs	B-Application
of	O
antiderivatives	B-Algorithm
of	O
a	O
given	O
function	O
are	O
vertical	B-Algorithm
translations	I-Algorithm
of	O
each	O
other	O
,	O
with	O
each	O
graph	O
's	O
vertical	O
location	O
depending	O
upon	O
the	O
value	O
.	O
</s>
<s>
More	O
generally	O
,	O
the	O
power	O
function	O
has	O
antiderivative	B-Algorithm
if	O
,	O
and	O
if	O
.	O
</s>
<s>
The	O
constant	O
is	O
the	O
initial	O
velocity	O
term	O
that	O
would	O
be	O
lost	O
upon	O
taking	O
the	O
derivative	B-Algorithm
of	O
velocity	O
,	O
because	O
the	O
derivative	B-Algorithm
of	O
a	O
constant	O
term	O
is	O
zero	O
.	O
</s>
<s>
This	O
same	O
pattern	O
applies	O
to	O
further	O
integrations	O
and	O
derivatives	B-Algorithm
of	O
motion	O
(	O
position	O
,	O
velocity	O
,	O
acceleration	O
,	O
and	O
so	O
on	O
)	O
.	O
</s>
<s>
Antiderivatives	B-Algorithm
can	O
be	O
used	O
to	O
compute	O
definite	B-Algorithm
integrals	I-Algorithm
,	O
using	O
the	O
fundamental	O
theorem	O
of	O
calculus	O
:	O
if	O
is	O
an	O
antiderivative	B-Algorithm
of	O
the	O
integrable	O
function	O
over	O
the	O
interval	O
,	O
then	O
:	O
</s>
<s>
Because	O
of	O
this	O
,	O
each	O
of	O
the	O
infinitely	O
many	O
antiderivatives	B-Algorithm
of	O
a	O
given	O
function	O
may	O
be	O
called	O
the	O
"	O
indefinite	B-Algorithm
integral	I-Algorithm
"	O
of	O
f	O
and	O
written	O
using	O
the	O
integral	O
symbol	O
with	O
no	O
bounds	O
:	O
</s>
<s>
If	O
is	O
an	O
antiderivative	B-Algorithm
of	O
,	O
and	O
the	O
function	O
is	O
defined	O
on	O
some	O
interval	O
,	O
then	O
every	O
other	O
antiderivative	B-Algorithm
of	O
differs	O
from	O
by	O
a	O
constant	O
:	O
there	O
exists	O
a	O
number	O
such	O
that	O
for	O
all	O
.	O
</s>
<s>
Every	O
continuous	O
function	O
has	O
an	O
antiderivative	B-Algorithm
,	O
and	O
one	O
antiderivative	B-Algorithm
is	O
given	O
by	O
the	O
definite	B-Algorithm
integral	I-Algorithm
of	O
with	O
variable	O
upper	O
boundary	O
:	O
</s>
<s>
Varying	O
the	O
lower	O
boundary	O
produces	O
other	O
antiderivatives	B-Algorithm
,	O
but	O
not	O
necessarily	O
all	O
possible	O
antiderivatives	B-Algorithm
.	O
</s>
<s>
There	O
are	O
many	O
functions	O
whose	O
antiderivatives	B-Algorithm
,	O
even	O
though	O
they	O
exist	O
,	O
cannot	O
be	O
expressed	O
in	O
terms	O
of	O
elementary	O
functions	O
(	O
like	O
polynomials	O
,	O
exponential	O
functions	O
,	O
logarithms	O
,	O
trigonometric	O
functions	O
,	O
inverse	O
trigonometric	O
functions	O
and	O
their	O
combinations	O
)	O
.	O
</s>
<s>
Finding	O
antiderivatives	B-Algorithm
of	O
elementary	O
functions	O
is	O
often	O
considerably	O
harder	O
than	O
finding	O
their	O
derivatives	B-Algorithm
(	O
indeed	O
,	O
there	O
is	O
no	O
pre-defined	O
method	O
for	O
computing	O
indefinite	B-Algorithm
integrals	I-Algorithm
)	O
.	O
</s>
<s>
For	O
some	O
elementary	O
functions	O
,	O
it	O
is	O
impossible	O
to	O
find	O
an	O
antiderivative	B-Algorithm
in	O
terms	O
of	O
other	O
elementary	O
functions	O
.	O
</s>
<s>
There	O
exist	O
many	O
properties	O
and	O
techniques	O
for	O
finding	O
antiderivatives	B-Algorithm
.	O
</s>
<s>
Inverse	O
function	O
integration	O
(	O
a	O
formula	O
that	O
expresses	O
the	O
antiderivative	B-Algorithm
of	O
the	O
inverse	O
of	O
an	O
invertible	O
and	O
continuous	O
function	O
,	O
in	O
terms	O
of	O
the	O
antiderivative	B-Algorithm
of	O
and	O
of	O
)	O
.	O
</s>
<s>
Computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
can	O
be	O
used	O
to	O
automate	O
some	O
or	O
all	O
of	O
the	O
work	O
involved	O
in	O
the	O
symbolic	O
techniques	O
above	O
,	O
which	O
is	O
particularly	O
useful	O
when	O
the	O
algebraic	O
manipulations	O
involved	O
are	O
very	O
complex	O
or	O
lengthy	O
.	O
</s>
<s>
Integrals	B-Algorithm
which	O
have	O
already	O
been	O
derived	O
can	O
be	O
looked	O
up	O
in	O
a	O
table	O
of	O
integrals	B-Algorithm
.	O
</s>
<s>
Non-continuous	O
functions	O
can	O
have	O
antiderivatives	B-Algorithm
.	O
</s>
<s>
Some	O
highly	O
pathological	O
functions	O
with	O
large	O
sets	O
of	O
discontinuities	O
may	O
nevertheless	O
have	O
antiderivatives	B-Algorithm
.	O
</s>
<s>
In	O
some	O
cases	O
,	O
the	O
antiderivatives	B-Algorithm
of	O
such	O
pathological	O
functions	O
may	O
be	O
found	O
by	O
Riemann	O
integration	O
,	O
while	O
in	O
other	O
cases	O
these	O
functions	O
are	O
not	O
Riemann	O
integrable	O
.	O
</s>
<s>
A	O
necessary	O
,	O
but	O
not	O
sufficient	O
,	O
condition	O
for	O
a	O
function	O
to	O
have	O
an	O
antiderivative	B-Algorithm
is	O
that	O
have	O
the	O
intermediate	O
value	O
property	O
.	O
</s>
<s>
Moreover	O
,	O
for	O
any	O
meagre	O
F-sigma	O
set	O
,	O
one	O
can	O
construct	O
some	O
function	O
having	O
an	O
antiderivative	B-Algorithm
,	O
which	O
has	O
the	O
given	O
set	O
as	O
its	O
set	O
of	O
discontinuities	O
.	O
</s>
<s>
If	O
has	O
an	O
antiderivative	B-Algorithm
,	O
is	O
bounded	O
on	O
closed	O
finite	O
subintervals	O
of	O
the	O
domain	O
and	O
has	O
a	O
set	O
of	O
discontinuities	O
of	O
Lebesgue	O
measure	O
0	O
,	O
then	O
an	O
antiderivative	B-Algorithm
may	O
be	O
found	O
by	O
integration	O
in	O
the	O
sense	O
of	O
Lebesgue	O
.	O
</s>
<s>
In	O
fact	O
,	O
using	O
more	O
powerful	O
integrals	B-Algorithm
like	O
the	O
Henstock	O
–	O
Kurzweil	O
integral	O
,	O
every	O
function	O
for	O
which	O
an	O
antiderivative	B-Algorithm
exists	O
is	O
integrable	O
,	O
and	O
its	O
general	B-Algorithm
integral	I-Algorithm
coincides	O
with	O
its	O
antiderivative	B-Algorithm
.	O
</s>
<s>
If	O
has	O
an	O
antiderivative	B-Algorithm
on	O
a	O
closed	O
interval	O
,	O
then	O
for	O
any	O
choice	O
of	O
partition	O
if	O
one	O
chooses	O
sample	O
points	O
as	O
specified	O
by	O
the	O
mean	B-Algorithm
value	I-Algorithm
theorem	I-Algorithm
,	O
then	O
the	O
corresponding	O
Riemann	O
sum	O
telescopes	O
to	O
the	O
value	O
.	O
</s>
