To solve the integral $\int e^{2x} \cos(e^x) \, dx$, we'll use substitution to simplify it.

Step 1: Substitution

Let $u = e^x$. Then, $du = e^x \, dx$. Notice that we can express $e^{2x}$ as $(e^x)^2 = u^2$, so the integral becomes:

$\int e^{2x} \cos(e^x) \, dx = \int u^2 \cos(u) \, \frac{du}{u}$

This simplifies to:

$\int u \cos(u) \, du$

Step 2: Integration by Parts

Now, we'll apply integration by parts to the integral $\int u \cos(u) \, du$. Let:

$v = u \quad \text{and} \quad dw = \cos(u) \, du$

Then, differentiate and integrate:

$dv = du \quad \text{and} \quad w = \sin(u)$

Using the integration by parts formula $\int v \, dw = vw - \int w \, dv$:

$\int u \cos(u) \, du = u \sin(u) - \int \sin(u) \, du$

Step 3: Simplify the Result

Integrate $\int \sin(u) \, du$:

$\int u \cos(u) \, du = u \sin(u) + \cos(u) + C$

Step 4: Substitute Back $u = e^x$

Finally, substitute $u = e^x$ back into the equation:

$\int e^{2x} \cos(e^x) \, dx = e^x \sin(e^x) + \cos(e^x) + C$

Final Answer

$\int e^{2x} \cos(e^x) \, dx = e^x \sin(e^x) + \cos(e^x) + C$

Would you like further details on any part of this solution? Here are some related questions:

1. What are the conditions for using substitution in integration?
2. How does integration by parts help in solving integrals with products?
3. What are the common trigonometric integrals?
4. How can substitution simplify integrals involving exponential functions?
5. What are some techniques to solve more complex trigonometric integrals?

Tip: Integration by parts is a powerful tool for integrals involving products of functions, especially when one function is easily differentiable and the other is easily integrable.