To integrate the function \( \frac{1}{1+x^2} \) over the interval \([0, 1]\), we'll proceed as follows:

### Step 1: Recognize the Integral Formula
The function \( \frac{1}{1+x^2} \) is a standard integrand, and its antiderivative is:

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

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

### Step 2: Apply the Definite Integral
We need to evaluate the definite integral from 0 to 1:

\[
\int_0^1 \frac{1}{1+x^2} \, dx = \left[\arctan(x)\right]_0^1
\]

### Step 3: Compute the Antiderivative at the Limits
Now, substitute the limits of integration into the antiderivative:

\[
\arctan(1) - \arctan(0)
\]

- \( \arctan(1) = \frac{\pi}{4} \)
- \( \arctan(0) = 0 \)

### Step 4: Final Answer
The result of the definite integral is:

\[
\frac{\pi}{4} - 0 = \frac{\pi}{4}
\]

Thus, the integral of \( \frac{1}{1+x^2} \) from 0 to 1 is \( \frac{\pi}{4} \).

Would you like further details or explanations on any step?

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1. What is the geometric interpretation of the arctan function?
2. How does the integral change if the limits are from \([0, a]\) where \(a > 1\)?
3. Can the integral be computed using a numerical method instead of analytically?
4. What are the properties of the arctan function that make this integral solvable?
5. How does this integral relate to inverse trigonometric functions?

**Tip:** In calculus, recognizing standard integrals can greatly simplify solving definite integrals. Familiarize yourself with common antiderivatives like arctan(x).

Explore the step-by-step process to compute the definite integral of the function 1/(1+x^2) over the interval [0, 1]. This problem utilizes key calculus concepts such as integration and the arctan function, and is suitable for high school students in Grades 11-12.

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