`int _(2) ^(3)(x)/( x ^(2) +1) dx = 1/2 log 2`. True or False.

Answer

To determine whether the integral \( \int \frac{x}{x^2 + 1} \, dx = \frac{1}{2} \log 2 \) is true or false, we will evaluate the integral step by step.

### Step 1: Identify the integral
We start with the integral:
\[ I = \int \frac{x}{x^2 + 1} \, dx \]

### Step 2: Use substitution
Let \( y = x^2 + 1 \). Then, the derivative \( dy \) is:
\[ dy = 2x \, dx \quad \Rightarrow \quad dx = \frac{dy}{2x} \]
Now, we can rewrite \( I \):
\[ I = \int \frac{x}{y} \cdot \frac{dy}{2x} = \frac{1}{2} \int \frac{1}{y} \, dy \]

### Step 3: Integrate
The integral of \( \frac{1}{y} \) is:
\[ \int \frac{1}{y} \, dy = \log |y| + C \]
Thus, we have:
\[ I = \frac{1}{2} \log |y| + C = \frac{1}{2} \log |x^2 + 1| + C \]

### Step 4: Evaluate the integral
Since \( x^2 + 1 \) is always positive, we can drop the absolute value:
\[ I = \frac{1}{2} \log (x^2 + 1) + C \]

### Step 5: Compare with the given statement
Now, we need to check if this result equals \( \frac{1}{2} \log 2 \). The expression \( \frac{1}{2} \log 2 \) is a constant. For the integral to equal \( \frac{1}{2} \log 2 \), we would need:
\[ \frac{1}{2} \log (x^2 + 1) + C = \frac{1}{2} \log 2 \]
This implies:
\[ \log (x^2 + 1) = \log 2 + C' \]
where \( C' = 2C \).

### Conclusion
The integral \( \int \frac{x}{x^2 + 1} \, dx \) evaluates to \( \frac{1}{2} \log (x^2 + 1) + C \), which is not equal to \( \frac{1}{2} \log 2 \) for all \( x \). Therefore, the statement is **False**.