what would the antiderivative of e^-2x be? and if the antiderivative of that is -1/2e^-2x, what would that antiderivative be?
To find the antiderivative of a function, you can use the power rule for integration, which states that if the derivative of a function is of the form f'(x) = e^(kx), then the antiderivative is given by F(x) = (1/k) * e^(kx).
In the case of e^(-2x), the derivative of this function is (-2) * e^(-2x). So to find the antiderivative, we can apply the power rule in reverse:
F(x) = (1/-2) * e^(-2x) = -1/2 * e^(-2x)
Now, if the antiderivative of -1/2 * e^(-2x) is given as a new function, let's call it G(x), then G(x) is the antiderivative of -1/2 * e^(-2x).
It's important to note that finding the antiderivative of an antiderivative simplifies to finding the original function.
Therefore, if the antiderivative of -1/2 * e^(-2x) is -1/2 * e^(-2x), the original function of G(x) is G(x) = -1/2 * e^(-2x).
In summary:
The antiderivative of e^(-2x) is -1/2 * e^(-2x).
The antiderivative of -1/2 * e^(-2x) is -1/2 * e^(-2x), which is the same as the original function.