`int( (log x)^(5) )/( x ) dx ` is equal to a) `(log x^(6))/( 6) + C` b) `( ( log x )^(6))/( 6) + C` c) `((log x )^(6))/( 3x^(2)) + C` d) none of these

Options

A. `(log x^(6))/( 6) + C`
B. `( ( log x )^(6))/( 6) + C`
C. `((log x )^(6))/( 3x^(2)) + C`
D. none of these

Answer

To solve the integral \( \int \frac{(\log x)^5}{x} \, dx \), we can use a substitution method. Here’s the step-by-step solution:

### Step 1: Substitution
Let \( t = \log x \). Then, the derivative of \( t \) with respect to \( x \) is:
\[ \frac{dt}{dx} = \frac{1}{x} \implies dt = \frac{1}{x} \, dx \implies dx = x \, dt = e^t \, dt \]

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ \int \frac{(\log x)^5}{x} \, dx = \int t^5 \, dt \]

### Step 3: Integrate
Now we integrate \( t^5 \):
\[ \int t^5 \, dt = \frac{t^6}{6} + C \]

### Step 4: Substitute Back
Now, substitute back \( t = \log x \):
\[ \frac{t^6}{6} + C = \frac{(\log x)^6}{6} + C \]

### Final Answer
Thus, the integral \( \int \frac{(\log x)^5}{x} \, dx \) is equal to:
\[ \frac{(\log x)^6}{6} + C \]

### Conclusion
The correct answer is option **b)** \( \frac{(\log x)^6}{6} + C \).

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