To solve the integral

\[
\int \frac{e^x \, dx}{e^x + 5}
\]

we can use the substitution method. Let's proceed with the following steps:

### Step 1: Choose a substitution
Let
\[
u = e^x + 5.
\]

Then, the derivative of \( u \) with respect to \( x \) is:
\[
\frac{du}{dx} = e^x \Rightarrow du = e^x \, dx.
\]

This substitution transforms the integral into:

\[
\int \frac{e^x \, dx}{e^x + 5} = \int \frac{du}{u}.
\]

### Step 2: Integrate with respect to \( u \)
Now we have a simpler integral:
\[
\int \frac{du}{u} = \ln |u| + C,
\]
where \( C \) is the constant of integration.

### Step 3: Substitute back in terms of \( x \)
Since \( u = e^x + 5 \), we can write the solution as:
\[
\ln |e^x + 5| + C.
\]

### Final Answer
\[
\int \frac{e^x \, dx}{e^x + 5} = \ln |e^x + 5| + C.
\]

Would you like a deeper explanation of any part of this solution?

Here are five related questions you may find interesting:
1. How would the approach differ if there was an extra constant multiplied with \( e^x \) in the denominator?
2. Can this integral be solved with other techniques like partial fractions?
3. What happens if we integrate \( \int \frac{e^{2x} \, dx}{e^x + 5} \)?
4. How does changing the constant in the denominator affect the integration result?
5. How would the solution change if we had \( \int \frac{e^{x} \, dx}{e^{2x} + 5} \) instead?

**Tip:** Substitution is a powerful method in integration; always look for expressions in the integrand that can simplify through derivatives!

Explore how to solve the integral \( \int \frac{e^x \, dx}{e^x + 5} \) using the substitution method. This step-by-step solution covers calculus concepts and the formula for integrating 1/u. Suitable for students in college calculus or advanced high school courses.

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