`int (sec^(2)(logx ))/(x)dx`

Answer

To solve the integral \( I = \int \frac{\sec^2(\log x)}{x} \, dx \), we will use substitution. Here’s a step-by-step solution:

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

### Step 2: Change of Variables
Substituting \( t = \log x \) into the integral, we have:
\[ I = \int \sec^2(t) \cdot \frac{1}{e^t} \cdot e^t \, dt = \int \sec^2(t) \, dt \]

### Step 3: Integrate
The integral of \( \sec^2(t) \) is a standard integral:
\[ \int \sec^2(t) \, dt = \tan(t) + C \]

### Step 4: Back Substitute
Now, we substitute back \( t = \log x \):
\[ I = \tan(\log x) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{\sec^2(\log x)}{x} \, dx = \tan(\log x) + C \]

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