To solve the integral of $e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) \, dx$, we can break it up into two separate integrals:

$\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) \, dx = \int e^x \frac{1}{x} \, dx - \int e^x \frac{1}{x^2} \, dx$

First Integral: $\int e^x \frac{1}{x} \, dx$

There is no elementary antiderivative for this term. The result of this integral is typically expressed using the Exponential Integral function, denoted as $\text{Ei}(x)$. Thus:

$\int e^x \frac{1}{x} \, dx = \text{Ei}(x)$

Second Integral: $\int e^x \frac{1}{x^2} \, dx$

We can solve this by using integration by parts. Let’s set:

• $u = \frac{1}{x^2}$, which means $du = -\frac{2}{x^3} dx$,
• $dv = e^x dx$, which gives $v = e^x$.

Now, applying the integration by parts formula: $\int u \, dv = uv - \int v \, du$ we get: $\int e^x \frac{1}{x^2} \, dx = e^x \frac{1}{x^2} - \int e^x \left( -\frac{2}{x^3} \right) \, dx$

This leads to the same type of integral with higher powers of $\frac{1}{x^n}$. However, these terms eventually tend toward more complex expressions involving the Exponential Integral function for non-elementary antiderivatives.

Final Solution

The solution to the integral can be written as: $\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) \, dx = \text{Ei}(x) + e^x \frac{1}{x^2} + \cdots$ where the remaining terms follow the same pattern.

Would you like to dive deeper into the solution, or have further questions?

Here are five follow-up questions that might interest you:

1. What is the Exponential Integral function?
2. Can we express non-elementary integrals in terms of special functions?
3. How does integration by parts help in simplifying integrals?
4. How are asymptotic expansions used to approximate solutions?
5. What are the practical applications of the Exponential Integral function?

Tip: Non-elementary integrals often arise in advanced calculus, and using special functions can greatly simplify expressing their solutions!