Question

find the antiderivative for each function when c equals 0.
a. ( f(x)=e^{7x} ) b. ( g(x)=e^{-8x} ) c. ( h(x)=e^{\frac{x}{9}} )
a. the antiderivative of ( e^{7x} ) is ( \frac{1}{7}e^{7x} ).
b. the antiderivative of ( e^{-8x} ) is ( -\frac{1}{8}e^{-8x} ).
c. the antiderivative of ( e^{\frac{x}{9}} ) is ( square ).

Explanation:

Step1: Recall the antiderivative formula

The antiderivative of \(e^{ax}\) is \(\frac{1}{a}e^{ax}+C\) (when \(a
eq0\)). For \(h(x) = e^{\frac{x}{9}}\), here \(a=\frac{1}{9}\).

Step2: Apply the formula

Using the formula \(\int e^{ax}dx=\frac{1}{a}e^{ax}+C\) with \(a = \frac{1}{9}\) and \(C = 0\), we have \(\int e^{\frac{x}{9}}dx=\frac{1}{\frac{1}{9}}e^{\frac{x}{9}}\).
Simplify \(\frac{1}{\frac{1}{9}}e^{\frac{x}{9}}\) to \(9e^{\frac{x}{9}}\).

Answer:

\(9e^{\frac{x}{9}}\)