If the integral `I=∫(-(sinx)/(x)-ln x cosx)dx=f(x)+C` (where, C is the constant of integration) and `f(e )=-sine`, then the number of natural numbers less than `[f((pi)/(6))]` is equal to (where `[.]` is the greatest integer function)

Answer

To solve the integral \( I = \int \left( -\frac{\sin x}{x} - \ln x \cos x \right) dx = f(x) + C \), where \( C \) is the constant of integration, and given that \( f(e) = -\sin e \), we need to find the number of natural numbers less than \( [f(\frac{\pi}{6})] \).

### Step-by-Step Solution:

1. **Separate the Integral**:
We can express the integral as:
\[ I = -\int \frac{\sin x}{x} \, dx - \int \ln x \cos x \, dx \]

2. **Integration by Parts**:
For the second integral \( \int \ln x \cos x \, dx \), we will use integration by parts. Let:
- \( u = \ln x \) ⇒ \( du = \frac{1}{x} \, dx \)
- \( dv = \cos x \, dx \) ⇒ \( v = \sin x \)

Applying integration by parts:
\[ \int \ln x \cos x \, dx = u v - \int v \, du = \ln x \sin x - \int \sin x \cdot \frac{1}{x} \, dx \]

3. **Combine the Integrals**:
Substitute back into the expression for \( I \):
\[ I = -\int \frac{\sin x}{x} \, dx - \left( \ln x \sin x - \int \frac{\sin x}{x} \, dx \right) \]
Simplifying gives:
\[ I = -\ln x \sin x + C \]

4. **Identify \( f(x) \)**:
From the integral, we have:
\[ f(x) = -\sin x \ln x \]

5. **Use the Given Condition**:
We know \( f(e) = -\sin e \). Let's verify:
\[ f(e) = -\sin e \ln e = -\sin e \cdot 1 = -\sin e \]
This confirms our expression for \( f(x) \).

6. **Evaluate \( f\left(\frac{\pi}{6}\right) \)**:
Now we need to calculate:
\[ f\left(\frac{\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) \ln\left(\frac{\pi}{6}\right) \]
We know \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \):
\[ f\left(\frac{\pi}{6}\right) = -\frac{1}{2} \ln\left(\frac{\pi}{6}\right) \]

7. **Approximate \( \ln\left(\frac{\pi}{6}\right) \)**:
Using \( \pi \approx 3.14 \):
\[ \frac{\pi}{6} \approx \frac{3.14}{6} \approx 0.5233 \]
Therefore, \( \ln\left(\frac{\pi}{6}\right) \) can be approximated:
\[ \ln(0.5233) \approx -0.644 \]
Hence,
\[ f\left(\frac{\pi}{6}\right) \approx -\frac{1}{2} \cdot (-0.644) \approx 0.322 \]

8. **Find the Greatest Integer**:
Now we apply the greatest integer function:
\[ [f\left(\frac{\pi}{6}\right)] = [0.322] = 0 \]

9. **Count Natural Numbers Less Than 0**:
The natural numbers less than 0 are none. Therefore, the answer is:
\[ \text{Number of natural numbers less than } [f\left(\frac{\pi}{6}\right)] = 0 \]

### Final Answer:
The number of natural numbers less than \( [f\left(\frac{\pi}{6}\right)] \) is \( \boxed{0} \).