`int_(1//e)^(e) (dx)/(x(log x)^(1//3))`

Answer

To solve the integral

\[ I = \int_{\frac{1}{e}}^{e} \frac{dx}{x (\log x)^{\frac{1}{3}}} \]

we will follow these steps:

### Step 1: Change of Variables
Let \( t = \log x \). Then, we have:

\[ dx = e^t dt \]

Also, since \( x = e^t \), we can rewrite \( \frac{dx}{x} \) as:

\[ \frac{dx}{x} = \frac{e^t dt}{e^t} = dt \]

### Step 2: Change the Limits of Integration
When \( x = \frac{1}{e} \):

\[ t = \log\left(\frac{1}{e}\right) = -1 \]

When \( x = e \):

\[ t = \log(e) = 1 \]

Thus, the limits of integration change from \( x = \frac{1}{e} \) to \( x = e \) into \( t = -1 \) to \( t = 1 \).

### Step 3: Rewrite the Integral
Now we can rewrite the integral in terms of \( t \):

\[ I = \int_{-1}^{1} \frac{dt}{t^{\frac{1}{3}}} \]

### Step 4: Simplify the Integral
The integral can be expressed as:

\[ I = \int_{-1}^{1} t^{-\frac{1}{3}} dt \]

### Step 5: Evaluate the Integral
This integral can be computed using the formula for the integral of a power function:

\[ \int t^{n} dt = \frac{t^{n+1}}{n+1} + C \quad \text{for } n \neq -1 \]

Here, \( n = -\frac{1}{3} \):

\[ I = \left[ \frac{t^{\frac{2}{3}}}{\frac{2}{3}} \right]_{-1}^{1} = \left[ \frac{3}{2} t^{\frac{2}{3}} \right]_{-1}^{1} \]

### Step 6: Calculate the Values at the Limits
Now we evaluate this from \( -1 \) to \( 1 \):

\[ I = \frac{3}{2} \left( 1^{\frac{2}{3}} - (-1)^{\frac{2}{3}} \right) \]

Since \( (-1)^{\frac{2}{3}} = 1 \):

\[ I = \frac{3}{2} \left( 1 - 1 \right) = \frac{3}{2} \cdot 0 = 0 \]

### Final Result
Thus, the value of the integral is:

\[ I = 0 \]

---