7–64. Integration review Evaluate the following integrals.
28. ∫ (3x + 1) / √(4 - x²) dx

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Recognize that the integral involves a square root in the denominator, specifically √(4 - x²). This suggests that a trigonometric substitution might be useful. Recall that for expressions of the form √(a² - x²), the substitution x = a sin(θ) is often effective.

Substitute x = 2sin(θ), which implies dx = 2cos(θ)dθ. Also, note that √(4 - x²) becomes √(4 - (2sin(θ))²) = √(4 - 4sin²(θ)) = 2cos(θ) using the Pythagorean identity 1 - sin²(θ) = cos²(θ).

Rewrite the integral in terms of θ using the substitution. The numerator (3x + 1) becomes 3(2sin(θ)) + 1 = 6sin(θ) + 1, and the denominator √(4 - x²) becomes 2cos(θ). The integral now becomes ∫ [(6sin(θ) + 1) / (2cos(θ))] * 2cos(θ)dθ.

Simplify the integral. The cos(θ) terms in the numerator and denominator cancel out, leaving ∫ (6sin(θ) + 1)dθ. Split this into two separate integrals: ∫ 6sin(θ)dθ + ∫ 1dθ.

Evaluate each integral. For ∫ 6sin(θ)dθ, use the fact that the integral of sin(θ) is -cos(θ). For ∫ 1dθ, the result is simply θ. After integrating, substitute back x = 2sin(θ) to return to the original variable x, using the relationship θ = arcsin(x/2).

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Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fractions. In this case, recognizing that the integral involves a rational function and a square root suggests that substitution may simplify the process.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots. By substituting a variable with a trigonometric function, such as x = 2sin(θ) for √(4 - x²), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals with expressions like √(a² - x²).

Definite vs. Indefinite Integrals

Definite integrals calculate the area under a curve between two limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is crucial for evaluating integrals correctly. In this problem, the integral is indefinite, meaning the result will include a constant term.

7–64. Integration review Evaluate the following integrals.28. ∫ (3x + 1) / √(4 - x²) dx