Question Stimulus :-
`int1/(x(logx)^2)dx=?`

Options

A. `1/(logx)+c`
B. `-1/(logx)+c`
C. log logx+c
D. `-log logx+c`

Answer

To solve the integral \( \int \frac{1}{x (\log x)^2} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \log x \). Then, we differentiate \( t \) with respect to \( x \):
\[ dt = \frac{1}{x} \, dx \quad \Rightarrow \quad dx = x \, dt = e^t \, dt \]
Now, we can express \( x \) in terms of \( t \):
\[ x = e^t \]

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ \int \frac{1}{x (\log x)^2} \, dx = \int \frac{1}{e^t t^2} \cdot e^t \, dt = \int \frac{1}{t^2} \, dt \]

### Step 3: Integrate
Now, we can integrate \( \frac{1}{t^2} \):
\[ \int \frac{1}{t^2} \, dt = -\frac{1}{t} + C \]

### Step 4: Substitute Back
Now, we substitute back \( t = \log x \):
\[ -\frac{1}{t} + C = -\frac{1}{\log x} + C \]

### Final Answer
Thus, the solution to the integral is:
\[ \int \frac{1}{x (\log x)^2} \, dx = -\frac{1}{\log x} + C \]

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