To solve the given integral:

$\int \frac{e^x}{x} \left( 1 + x \ln x \right) dx$

We will approach this step by step:

Step 1: Break Down the Expression

The integral can be separated into two parts: $\int \frac{e^x}{x} \left( 1 + x \ln x \right) dx = \int \frac{e^x}{x} dx + \int e^x \ln x \, dx.$

Step 2: Solve Each Term

Term 1: $\int \frac{e^x}{x} dx$

The integral $\int \frac{e^x}{x} dx$ does not have a solution in terms of elementary functions. This is known as the exponential integral and is typically expressed using the special function $\text{Ei}(x)$.

Term 2: $\int e^x \ln x \, dx$

This requires integration by parts:

• Let $u = \ln x$, so that $du = \frac{1}{x} dx$.
• Let $dv = e^x dx$, so that $v = e^x$.

Using the integration by parts formula: $\int u \, dv = uv - \int v \, du,$ we get: $\int e^x \ln x \, dx = e^x \ln x - \int e^x \cdot \frac{1}{x} dx.$

Simplified Result

Thus, the integral becomes: $\int \frac{e^x}{x} \left( 1 + x \ln x \right) dx = \text{Ei}(x) + \left( e^x \ln x - \text{Ei}(x) \right).$

Combining terms, the final solution is: $\int \frac{e^x}{x} \left( 1 + x \ln x \right) dx = e^x \ln x.$

Let me know if you'd like further clarifications or step-by-step details!

Related Questions:

1. What is the definition and significance of the exponential integral function, $\text{Ei}(x)$?
2. How do we apply integration by parts for functions involving logarithms and exponentials?
3. Can $\int \frac{e^x}{x} dx$ be solved numerically? If so, how?
4. What are the limitations of expressing certain integrals in terms of elementary functions?
5. How does the substitution method compare to integration by parts in solving complex integrals?

Tip:

Always check if a special function or numerical approach is required for integrals involving expressions like $\frac{e^x}{x}$. This avoids unnecessary complications in manual computation.