Step 1 :We are asked to find the antiderivative of the function \(f(x) = e^{-7x}\).
Step 2 :The antiderivative of \(e^{ax}\), where \(a\) is a constant, is \(\frac{1}{a}e^{ax}\). Therefore, the antiderivative of \(e^{-7x}\) should be \(-\frac{1}{7}e^{-7x}\).
Step 3 :However, the antiderivative is not unique. The general antiderivative of a function includes a constant of integration, usually denoted as \(C\). Therefore, the general antiderivative of \(e^{-7x}\) should be \(-\frac{1}{7}e^{-7x} + C\).
Step 4 :Final Answer: The general antiderivative of the function \(f(x) = e^{-7x}\) is \(\boxed{-\frac{1}{7}e^{-7x} + C}\).