`int e^(f(x)) dx = (x^(6))/(6) +C`, find f(x).

Answer

To solve the equation \( \int e^{f(x)} \, dx = \frac{x^6}{6} + C \) and find \( f(x) \), we will follow these steps:

### Step 1: Differentiate both sides
We start by differentiating both sides of the equation with respect to \( x \).

\[ \frac{d}{dx} \left( \int e^{f(x)} \, dx \right) = \frac{d}{dx} \left( \frac{x^6}{6} + C \right) \]

### Step 2: Apply the Fundamental Theorem of Calculus
According to the Fundamental Theorem of Calculus, the derivative of an integral is the integrand itself. Thus, we have:

\[ e^{f(x)} = x^5 \]

### Step 3: Take the natural logarithm of both sides
To isolate \( f(x) \), we take the natural logarithm (logarithm base \( e \)) of both sides:

\[ \ln(e^{f(x)}) = \ln(x^5) \]

### Step 4: Simplify using properties of logarithms
Using the property of logarithms that \( \ln(e^a) = a \) and \( \ln(a^b) = b \ln(a) \), we can simplify:

\[ f(x) = 5 \ln(x) \]

### Final Result
Thus, the function \( f(x) \) is:

\[ f(x) = 5 \ln(x) \]

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