Evaluate the Integral

Problem

(∫_−1^1)(1/(1+y2)*d(y))

Solution

1. Identify the antiderivative of the integrand 1/(1+y2) which is the inverse tangent function arctan(y)

2. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits of integration.

(∫_−1^1)(1/(1+y2)*d(y))=[arctan(y)]1(−1)

3. Substitute the limits into the expression.

arctan(1)−arctan(−1)

4. Evaluate the trigonometric values.

π/4−(−π/4)

5. Simplify the final result.

π/4+π/4=π/2

Final Answer

(∫_−1^1)(1/(1+y2)*d(y))=π/2

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