Evaluate : `int_(1)^(2) (x )/(1+x^(2))dx `

Answer

To evaluate the integral \( I = \int_{1}^{2} \frac{x}{1+x^2} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = 1 + x^2 \). Then, differentiate both sides with respect to \( x \):
\[ dt = 2x \, dx \quad \Rightarrow \quad x \, dx = \frac{dt}{2} \]

### Step 2: Change the limits of integration
Now we need to change the limits of integration according to our substitution:
- When \( x = 1 \):
\[ t = 1 + 1^2 = 2 \]
- When \( x = 2 \):
\[ t = 1 + 2^2 = 5 \]

### Step 3: Rewrite the integral
Now we can rewrite the integral in terms of \( t \):
\[ I = \int_{2}^{5} \frac{1}{t} \cdot \frac{dt}{2} \]
This simplifies to:
\[ I = \frac{1}{2} \int_{2}^{5} \frac{1}{t} \, dt \]

### Step 4: Evaluate the integral
The integral \( \int \frac{1}{t} \, dt \) is \( \log |t| \). Therefore:
\[ I = \frac{1}{2} \left[ \log t \right]_{2}^{5} \]
Calculating this gives:
\[ I = \frac{1}{2} \left( \log 5 - \log 2 \right) \]

### Step 5: Simplify the result
Using the properties of logarithms, we can combine the logs:
\[ I = \frac{1}{2} \log \left( \frac{5}{2} \right) \]

### Final Answer
Thus, the value of the integral is:
\[ I = \frac{1}{2} \log \left( \frac{5}{2} \right) \]

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