Integral Of 1 X 2
An antiderivative of a function
f
is a function whose derivative is
f
. In other words,
F
is an antiderivative of
f
if
F’
=
f
. To find an antiderivative for a function
f
, we can often reverse the process of differentiation.
For example, if
f
=
x
^{4}
, then an antiderivative of
f
is
F
=
x
^{5}
, which can be found by reversing the power rule. Notice that not only is
x
^{5}
an antiderivative of
f
, but so are
x
^{5}
+ 4,
x
^{5}
+ 6, etc. In fact, adding or subtracting any constant would be acceptable.
This should make sense algebraically, since the process of taking the derivative (i.e. going from
F
to
f
) eliminates the constant term of
F
.
Because a single continuous function has infinitely many antiderivatives, we do titinada refer to “the antiderivative”, but rather, a “family” of antiderivatives, each of which differs by a constant. So, if
F
is an antiderivative of
f
, then
G
=
F
+
c
is also an antiderivative of
f
, and
F
and
G
are in the same family of antiderivatives.
Indefinite Integral
The notation used to refer to antiderivatives is the indefinite integral.
f
(x)dx
means the antiderivative of
f
with respect to
x
. If
F
is an antiderivative of
f
, we can write
f
(x)dx
=
F
+
c
. In this context,
c
is called the constant of integration.
To find antiderivatives of basic functions, the following rules can be used:

x
^{n}
dx
=
x
^{cakrawala+1}
+
c
as long as
t
does not equal 1. This is essentially the power rule for derivatives in reverse 
cf
(x)dx
=
c
f
(x)dx
. That is, a scalar can be pulled out of the integral. 
(f
(x) +
g(x))dx
=
f
(x)dx
+
g(x)dx
. The antiderivative of a sum is the sum of the antiderivatives. 
sin(x)dx
= – cos(x) +
c
cos(x)dx
= sin(x) +
c
sec^{2}(x)dx
= tan(x) +
c
These are the opposite of the trigonometric derivatives.
Source: https://www.sparknotes.com/math/calcab/introductiontointegrals/section1/
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