Anti Derivative of 2^x
Antiderivative 2^x
Anti derivative of the function can be understood as if there is a function `f' then anti
derivative of function `f' is the function
`F' which has the derivative `f'. Sometimes anti derivative is also known as primitive
integral or indefinite integral.
Anti derivative or anti differentiation is defined as the process that is same as
integration and opposite to differentiation and it will give the definite integral of a
function for a given interval .
The obtained integral is same as the anti derivative of function calculated for end
points of the given interval.
Antiderivative 2^x: Let us take an example to understand this concept, if a function f
= 2x then it is read as Antiderivative 2 to the power `x' is solved as follows:
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If the above defined function is denoted as f (x) = x2, then antiderivative of the
function `f' is another function F (x) = x3 / 3.
As we know the derivative of the constant value is zero then there are infinite number
of antiderivatives that are describe as (x 3 / 3) + c, where `c' is the constant value and
value will vary as (x3 / 3) + 4 or (x3 / 3) - 6 or (x3 / 3) - 67 etc. so antiderivative of the
function is infinite .
Use of antiderivative in real life, it is used when we have value of acceleration and by
using the anti derivative we can obtain the velocity and constant with it and this
constant define the initial velocity that is different on different time intervals .
If we calculate the antiderivative of the function f = 2x, it will be solved as, we know e
x and ln (x) are opposite or inverse functions and they are define as x = eln (x).
So if we calculate the anti derivative of the above function d (2 x) / d x = d e ln (x) / d x
.
Put the value from above:
d (e ln (x)) / d x = ln (2)x .
If d (2x) / d x = ln (2) 2x.
So the anti derivative of the given function 2x is equals to 1 / ln (2) 2x.
In general way the derivative of function px is ln (p) px and anti derivative of the
function is.
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(1 / ln (p) p x).
There is also another way to solve the function f (x) = 2x directly as in general
f (p x) = px * ln (p) where `p' is a constant .
So if we assume ax is a variable that is define as v = 2x then
x = (1 / ln 2x) * ln v,
or d x = (1 / ln 2) * 1 / v * d v.
And further we solve it same as before. This is how we find the antideirvative of a
function f(x) = 2x that is opposite operation of differentiation or we can it is the
process of integration.


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