whats the antiderivitive?

Question

of 1/(2^ x)

Answer 1

To use the antiderivative method you must find the derivative i.e. d/dx[1/(2^x)].
d/dx[1/(2^x)] = d/dx[(1/2)^x]

We know d/dx(a^x) = (ln|a|)*(a^x).
This can be written as
[1/(ln|a|)]*d/dx(a^x) = a^x
or d/dx[(a^x)/(ln|a|)] = a^x

The whole part inside the bracket after d/dx is called the antiderivative of the right side.
In our case, a = 1/2
So the reqd. antiderivative is
(1/2)^x/(ln|1/2|)
= (1/2)^x/(-ln2)
= - 1/[(2^x)*ln(2)]

Note that ln(2) means natural logarithm with respect to base 'e'.

Answer 2

The antiderivative of 1/(2^ x) or
integral[dx/(2^ x)] = integral[2^(-x)dx] = [-2^(-x)]/ln(2) + c, c is a constant.

Answer 3

1/2^x = 2^-x

Rule:
derivitive of a^x= a^x multiplied by the natuaral log of a


so......
Dx 2^-x =2^-x Ln 2
=2^-x*0.693