The given integral is:

$\int \frac{e^x (1+x)}{\cos^2(x e^x)} \, dx$

To solve this integral, let's break it down step by step:

Step 1: Substitution

We notice that there is a complex function $x e^x$ in the cosine term. A good approach is to try a substitution to simplify the integral.

Let: $u = x e^x$ Then, to differentiate $u$ with respect to $x$:

$\frac{du}{dx} = e^x + x e^x = e^x (1 + x)$

Thus, $du = e^x (1+x) \, dx$, which means that the numerator becomes simply $du$.

Step 2: Substituting into the integral

Now, substitute $u$ and $du$ into the integral:

$\int \frac{du}{\cos^2(u)}$

This integral is a standard form:

$\int \sec^2(u) \, du$

Step 3: Integrating

We know that:

$\int \sec^2(u) \, du = \tan(u) + C$

Step 4: Substituting back $u$

Now, substitute back $u = x e^x$ to get the final solution:

$\tan(x e^x) + C$

Thus, the solution to the integral is:

$\tan(x e^x) + C$

Would you like further details or clarification on any of these steps?

Here are 5 related questions for further exploration:

1. How do you recognize when substitution is a good method for solving integrals?
2. What are other standard integrals related to trigonometric functions?
3. Can this method be applied to other forms of integrals with exponential and trigonometric combinations?
4. How does the behavior of the function $\tan(x e^x)$ differ from simpler functions like $\tan(x)$?
5. What would happen if there was a different power on the cosine term, e.g., $\cos^n(x e^x)$?

Tip: When working with exponential functions inside trigonometric terms, substitution is often helpful to simplify the argument of the trigonometric function.