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?

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.