Evaluate the Integral integral of 1/(4+x^2) with respect to x

Problem

(∫_^)(1/(4+x^2)*d(x))

Solution

1. Identify the form of the integral as a standard inverse trigonometric integral, specifically the form (∫_^)(1/(a^2+u^2)*d(u))

2. Determine the constant a by rewriting the denominator as 2^2+x^2 which gives a=2

3. Apply the formula for the arctangent integral, which states (∫_^)(1/(a^2+x^2)*d(x))=1/a*arctan(x/a)+C

4. Substitute a=2 into the formula to find the final result.

Final Answer

(∫_^)(1/(4+x^2)*d(x))=1/2*arctan(x/2)+C

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