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

Answer

To solve the integral

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

we will follow these steps:

### Step 1: Substitution
Let \( t = \log x \). Then, the differential \( dx \) can be expressed in terms of \( dt \):
\[ dx = e^t dt \]
Also, since \( x = e^t \), we have:
\[ \frac{dx}{x} = dt \]

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

### Step 3: Recognize the Standard Integral
The integral \( \int \frac{dt}{1 + t^2} \) is a standard integral, which evaluates to:
\[ \int \frac{dt}{1 + t^2} = \tan^{-1}(t) + C \]

### Step 4: Substitute Back
Since we made the substitution \( t = \log x \), we substitute back to get:
\[ I = \tan^{-1}(\log x) + C \]

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

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