https://file.aiquickdraw.com/mathbot/file/1726422227549ax4cvnvf.jpg

Let's solve the integrals shown in the image.

### Problem 63:
\[
\int_{-1}^{1} \frac{1}{1 + x^2} \, dx
\]
This is a standard integral, which is the integral of the arctangent function. The solution is:

\[
\int \frac{1}{1 + x^2} \, dx = \tan^{-1}(x) + C
\]

Applying the limits of integration:

\[
\int_{-1}^{1} \frac{1}{1 + x^2} \, dx = \left[ \tan^{-1}(x) \right]_{-1}^{1}
\]

Evaluating this:

\[
= \tan^{-1}(1) - \tan^{-1}(-1)
\]

Since \(\tan^{-1}(1) = \frac{\pi}{4}\) and \(\tan^{-1}(-1) = -\frac{\pi}{4}\), we get:

\[
\frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2}
\]

Thus, the result is:
\[
\boxed{\frac{\pi}{2}}
\]

---

### Problem 66:
\[
\int \frac{e^x}{1 + e^{2x}} \, dx
\]

For this integral, we can use a substitution. Let \( u = e^x \), which gives \( du = e^x \, dx \). This transforms the integral into:

\[
\int \frac{1}{1 + u^2} \, du
\]

This is again the standard arctangent integral:

\[
\int \frac{1}{1 + u^2} \, du = \tan^{-1}(u) + C
\]

Substituting back \( u = e^x \):

\[
\int \frac{e^x}{1 + e^{2x}} \, dx = \tan^{-1}(e^x) + C
\]

Thus, the result is:
\[
\boxed{\tan^{-1}(e^x) + C}
\]

Would you like further details or explanations?

Here are 5 related questions:
1. How can you use trigonometric substitutions for integrals like these?
2. What other methods besides substitution might work for problem 66?
3. How does symmetry help when integrating even functions over symmetric intervals?
4. Can you generalize the result of \(\int \frac{1}{1 + x^2} dx\) for other intervals?
5. What are some common integrals involving exponential functions?

**Tip**: When dealing with integrals involving expressions like \( 1 + x^2 \) or \( 1 + e^{2x} \), think of the arctangent function or trigonometric identities for easier solutions.

Solve the integrals: 1) \( \int_{-1}^{1} \frac{1}{1 + x^2} \, dx \) 2) \( \int \frac{e^x}{1 + e^{2x}} \, dx \)

Learn how to solve two integrals involving the arctangent function and exponential substitution. Detailed solutions show how to apply integral calculus techniques, such as the substitution method and inverse trigonometric function properties. Suitable for advanced high school or undergraduate students.

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