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The given problem asks for the evaluation of the integral:

\[
\int e^{2x} \sqrt{1 - e^{2x}} \, dx
\]
with the substitution \( u = 1 - e^{2x} \).

Let's go step by step to solve this:

### Step 1: Perform the substitution
We are given \( u = 1 - e^{2x} \). Now, differentiate both sides with respect to \( x \) to find \( du \):

\[
du = -2e^{2x} \, dx \quad \Rightarrow \quad \frac{du}{-2} = e^{2x} \, dx
\]

### Step 2: Substitute in the integral
Substitute \( e^{2x} \, dx \) with \( \frac{du}{-2} \) and \( 1 - e^{2x} \) with \( u \). The integral becomes:

\[
\int e^{2x} \sqrt{1 - e^{2x}} \, dx = \int \sqrt{u} \cdot \frac{du}{-2}
\]

### Step 3: Simplify the integral
This simplifies to:

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

### Step 4: Integrate
Now integrate:

\[
-\frac{1}{2} \int u^{1/2} \, du = -\frac{1}{2} \cdot \frac{2}{3} u^{3/2} = -\frac{1}{3} u^{3/2}
\]

### Step 5: Substitute back \( u \)
Finally, substitute \( u = 1 - e^{2x} \) back into the expression:

\[
-\frac{1}{3} \left( 1 - e^{2x} \right)^{3/2} + C
\]

### Final Answer:
The solution to the integral is:

\[
-\frac{1}{3} \left( 1 - e^{2x} \right)^{3/2} + C
\]

Would you like further details on any part of this process?

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Here are 5 related questions you could explore:
1. How does substitution simplify complex integrals in general?
2. Can this method be used for integrals with different exponents in \( e^x \)?
3. What is the geometric interpretation of an integral like this?
4. How does integration by substitution compare to other methods like integration by parts?
5. What are common mistakes to avoid when using substitution in integration?

**Tip:** Always differentiate your substitution to ensure you accurately replace \( dx \) in the integral.

Integrate using the indicated substitution: ∫ e^2x √(1 - e^2x) dx, with u = 1 - e^2x.

Learn how to solve the integral ∫ e^2x √(1 - e^2x) dx using substitution with u = 1 - e^2x. The step-by-step solution includes substitution, simplification, and integration techniques for students in advanced calculus.

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