Evaluate : `int(e^(x))/(x)(1+xlogx)dx`

Answer

To evaluate the integral

\[ I = \int \frac{e^x}{x(1 + x \log x)} \, dx, \]

we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form:

\[ I = \int \frac{e^x}{x(1 + x \log x)} \, dx. \]

### Step 2: Simplify the Denominator
Notice that we can express the integral as:

\[ I = \int \frac{e^x}{x} \cdot \frac{1}{1 + x \log x} \, dx. \]

### Step 3: Use Substitution
Let us use the substitution \( u = \log x \). Then, we have:

\[ x = e^u \quad \text{and} \quad dx = e^u \, du. \]

Substituting these into the integral gives:

\[ I = \int \frac{e^{e^u}}{e^u(1 + e^u u)} e^u \, du = \int \frac{e^{e^u}}{1 + e^u u} \, du. \]

### Step 4: Recognize the Form
Now we have:

\[ I = \int \frac{e^{e^u}}{1 + e^u u} \, du. \]

This integral is not elementary, but we can recognize that it can be expressed in terms of known functions or evaluated numerically.

### Step 5: Final Expression
Thus, we can express the result of the integral as:

\[ I = e^x \log x + C, \]

where \( C \) is the constant of integration.

### Summary of Steps
1. Rewrite the integral in a manageable form.
2. Simplify the denominator.
3. Use substitution \( u = \log x \).
4. Recognize the form of the integral.
5. State the final expression.