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The given integral is:

\[
\int_0^1 \frac{e^{2t}}{(3 + e^{2t})^2} \, dt
\]

### Solution:

We can approach this using substitution. Let's introduce a substitution to simplify the expression.

Let:
\[
u = 3 + e^{2t}
\]
Then,
\[
\frac{du}{dt} = 2e^{2t}
\quad \text{or} \quad du = 2e^{2t} dt
\]
Thus,
\[
\frac{e^{2t}}{(3 + e^{2t})^2} dt = \frac{1}{2} \cdot \frac{du}{u^2}
\]

Now, let's change the limits of integration according to the substitution:
- When \(t = 0\), \(u = 3 + e^{0} = 4\)
- When \(t = 1\), \(u = 3 + e^{2} = 3 + e^2\)

Therefore, the integral becomes:
\[
\frac{1}{2} \int_4^{3 + e^2} \frac{du}{u^2}
\]

The integral of \(\frac{1}{u^2}\) is:
\[
\int \frac{du}{u^2} = -\frac{1}{u}
\]

So, the integral evaluates to:
\[
-\frac{1}{2} \left[ \frac{1}{u} \right]_4^{3 + e^2}
\]
\[
= -\frac{1}{2} \left( \frac{1}{3 + e^2} - \frac{1}{4} \right)
\]

Simplifying:
\[
= \frac{1}{2} \left( \frac{1}{4} - \frac{1}{3 + e^2} \right)
\]

This is the final result in simplified form.

Would you like to go over any step in more detail, or have questions on the concepts used here?

### Further Exploration:
1. What other substitution techniques can simplify integrals?
2. How does the choice of substitution impact the limits of integration?
3. Can this integral be evaluated using numerical methods?
4. What if the exponent in \(e^{2t}\) were different—how would the solution change?
5. How does integrating with respect to \(e^t\) differ from \(e^{2t}\)?

**Tip:** Substitution is one of the most powerful methods in integration for reducing complex expressions to simpler forms!

Evaluate the integral from 0 to 1 of e^(2t) over (3 + e^(2t))^2 dt.

This calculus problem involves evaluating an integral using the substitution method. The integral is ∫(e^(2t)/(3 + e^(2t))^2) dt from 0 to 1, and the solution includes applying a substitution, adjusting limits of integration, and simplifying the result.

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