Evaluate the integrals in Exercises 1–14.
∫ dx / (8 + 2x²) from 0 to 2

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Identify the integral to be evaluated: \(\displaystyle \int_0^2 \frac{dx}{8 + 2x^2}\).

Factor out constants from the denominator to simplify the integral: rewrite the denominator as \(2(4 + x^2)\), so the integral becomes \(\displaystyle \int_0^2 \frac{dx}{2(4 + x^2)} = \frac{1}{2} \int_0^2 \frac{dx}{4 + x^2}\).

Recognize the integral form \(\int \frac{dx}{a^2 + x^2}\), which has the antiderivative \(\frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\). Here, \(a^2 = 4\), so \(a = 2\).

Apply the antiderivative formula to the integral: \(\frac{1}{2} \int_0^2 \frac{dx}{4 + x^2} = \frac{1}{2} \left[ \frac{1}{2} \arctan\left( \frac{x}{2} \right) \right]_0^2\).

Evaluate the definite integral by substituting the limits \(x=2\) and \(x=0\) into the antiderivative expression and subtracting: \(\frac{1}{4} \left( \arctan(1) - \arctan(0) \right)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the lower and upper bounds. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

Integration of Rational Functions

Integrating rational functions often requires recognizing standard integral forms or using substitution. For integrals involving expressions like 1/(a + bx²), the result typically involves inverse trigonometric functions such as arctangent, especially when the denominator is a sum of squares.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more familiar form. For example, substituting u = x√(coefficient) can help rewrite the integral into a standard arctangent form, making it easier to evaluate.

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