https://file.aiquickdraw.com/mathbot/file/17304701984552zfd35y0.jpg

The integral in the image is:

\[
\int \frac{e^x}{1 + e^{2x}} \, dx
\]

To solve this, we can use a substitution method.

### Solution

1. **Substitute**: Let \( u = e^x \). Then \( du = e^x \, dx \) or \( dx = \frac{du}{e^x} = \frac{du}{u} \).

2. **Rewrite the Integral**:
   Substitute \( u = e^x \) into the integral. The expression becomes:
   \[
   \int \frac{u}{1 + u^2} \cdot \frac{du}{u} = \int \frac{1}{1 + u^2} \, du
   \]

3. **Integrate**:
   Now, we have a standard integral:
   \[
   \int \frac{1}{1 + u^2} \, du = \arctan(u) + C
   \]

4. **Substitute Back**:
   Recall that \( u = e^x \), so:
   \[
   \int \frac{e^x}{1 + e^{2x}} \, dx = \arctan(e^x) + C
   \]

### Final Answer
\[
\int \frac{e^x}{1 + e^{2x}} \, dx = \arctan(e^x) + C
\]

Would you like further details on any step, or have any questions?

### Related Questions:
1. How can we solve integrals with other exponential substitutions?
2. What are other methods to integrate functions similar to this?
3. How does the substitution method apply to trigonometric integrals?
4. Can integration by parts be applied to exponential integrals?
5. What is the general form of the integral of \(\frac{e^{ax}}{1 + e^{2ax}}\)?

### Tip:
When dealing with an exponential function in the denominator, substitution with \( u = e^x \) often simplifies the integral significantly.

Evaluate the integral \( \int \frac{e^x}{1 + e^{2x}} \, dx \)

Learn how to evaluate the integral \( \int \frac{e^x}{1 + e^{2x}} \, dx \) using substitution. This solution applies the substitution \( u = e^x \) to simplify the expression, followed by integration of \( \frac{1}{1 + u^2} \), yielding \( \arctan(e^x) + C \). Suitable for high school students studying calculus.

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