https://file.aiquickdraw.com/mathbot/file/1726472922694z0iviw50.jpg

The integral given in the image is:

\[
\int \frac{e^x}{7e^x + 6} \, dx
\]

To solve this, we will use the substitution method.

1. **Substitution:**
   Let \( u = 7e^x + 6 \). Then, the derivative of \( u \) with respect to \( x \) is:
   \[
   \frac{du}{dx} = 7e^x
   \]
   So, \( du = 7e^x dx \), or \( \frac{du}{7} = e^x dx \).

2. **Rewriting the integral:**
   The integral becomes:
   \[
   \int \frac{e^x}{7e^x + 6} dx = \frac{1}{7} \int \frac{du}{u}
   \]

3. **Solving the integral:**
   The integral of \( \frac{1}{u} \) is \( \ln |u| \), so:
   \[
   \frac{1}{7} \ln |u| + C
   \]

4. **Substitute \( u \) back:**
   \[
   \frac{1}{7} \ln |7e^x + 6| + C
   \]

Thus, the solution is:

\[
\frac{1}{7} \ln(7e^x + 6) + C
\]

The correct answer is **option E**.

Would you like more details or further clarifications? Here are some related questions:

1. What is the substitution rule in integration?
2. How do you integrate functions involving exponentials?
3. What is the difference between indefinite and definite integrals?
4. How can you check your answer to an indefinite integral?
5. What are common mistakes when using substitution?

**Tip:** Always check if the function inside the logarithm requires absolute value bars for cases where the argument can be negative.

Evaluate the indefinite integral ∫ e^x / (7e^x + 6) dx.

Learn how to solve the indefinite integral ∫ e^x / (7e^x + 6) dx using the substitution method. This detailed solution covers key calculus concepts such as integration and the substitution rule, suitable for students in grades 11-12.

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