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:



  1. 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


  2. cf
    (x)dx
    =
    c

    f
    (x)dx
    . That is, a scalar can be pulled out of the integral.

  3. (f
    (x) +
    g(x))dx
    =

    f
    (x)dx
    +

    g(x)dx
    . The antiderivative of a sum is the sum of the antiderivatives.

  4. sin(x)dx
    = – cos(x) +
    c



    cos(x)dx
    = sin(x) +
    c



    sec2(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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