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!