`int_(0)^(oo)((ln(1+x^(2)))/(1+x^(2)))dx`.

Answer

To solve the integral

\[ I = \int_{0}^{\infty} \frac{\ln(1+x^2)}{1+x^2} \, dx, \]

we will use the substitution \( x = \tan \theta \). This gives us \( dx = \sec^2 \theta \, d\theta \) and transforms the limits of integration: when \( x = 0 \), \( \theta = 0 \) and when \( x = \infty \), \( \theta = \frac{\pi}{2} \).

### Step 1: Change of Variables
Substituting \( x = \tan \theta \):

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\ln(1+\tan^2 \theta)}{1+\tan^2 \theta} \sec^2 \theta \, d\theta. \]

Using the identity \( 1 + \tan^2 \theta = \sec^2 \theta \), we can simplify the integral:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\ln(\sec^2 \theta)}{\sec^2 \theta} \sec^2 \theta \, d\theta = \int_{0}^{\frac{\pi}{2}} \ln(\sec^2 \theta) \, d\theta. \]

### Step 2: Simplifying the Integral
We know that \( \ln(\sec^2 \theta) = 2 \ln(\sec \theta) \), so we can rewrite the integral:

\[ I = 2 \int_{0}^{\frac{\pi}{2}} \ln(\sec \theta) \, d\theta. \]

### Step 3: Evaluating the Integral
The integral \( \int_{0}^{\frac{\pi}{2}} \ln(\sec \theta) \, d\theta \) can be evaluated using the known result:

\[ \int_{0}^{\frac{\pi}{2}} \ln(\sec \theta) \, d\theta = -\frac{\pi}{2} \ln(2). \]

Thus,

\[ I = 2 \left(-\frac{\pi}{2} \ln(2)\right) = -\pi \ln(2). \]

### Final Result
Therefore, the value of the integral is:

\[ \int_{0}^{\infty} \frac{\ln(1+x^2)}{1+x^2} \, dx = -\pi \ln(2). \]

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