`int(dx)/(x(1+(logx)^(2)))` equals

Answer

To solve the integral

\[ I = \int \frac{dx}{x(1 + (\log x)^2)}, \]

we can follow these steps:

### Step 1: Substitution
Let \( t = \log x \). Then, the derivative of \( t \) with respect to \( x \) is

\[ \frac{dt}{dx} = \frac{1}{x} \implies dx = x \, dt = e^t \, dt. \]

### Step 2: Change of Variables
Substituting \( x = e^t \) into the integral gives us:

\[ I = \int \frac{e^t \, dt}{e^t(1 + t^2)} = \int \frac{dt}{1 + t^2}. \]

### Step 3: Integrate
The integral

\[ \int \frac{dt}{1 + t^2} \]

is a standard integral that equals

\[ \tan^{-1}(t) + C, \]

where \( C \) is the constant of integration.

### Step 4: Back Substitute
Now, substituting back \( t = \log x \):

\[ I = \tan^{-1}(\log x) + C. \]

### Final Answer
Thus, the final answer is

\[ \int \frac{dx}{x(1 + (\log x)^2)} = \tan^{-1}(\log x) + C. \]

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