Evaluate the Integral integral of 1/(x^2+16) with respect to x

Problem

(∫_^)(1/(x2+16)*d(x))

Solution

1. Identify the form of the integral as a standard inverse trigonometric integral, specifically the form (∫_^)(1/(x2+a2)*d(x))

2. Determine the value of the constant a by recognizing that 16=4 so a=4

3. Apply the formula for the arctangent integral, which states (∫_^)(1/(x2+a2)*d(x))=1/a*arctan(x/a)+C

4. Substitute a=4 into the formula to obtain the final result.

Final Answer

(∫_^)(1/(x2+16)*d(x))=1/4*arctan(x/4)+C

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