2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
18. ∫ (from 0 to √2) (x + 1)/(3x² + 6) dx

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Step 1: Simplify the integrand. Start by factoring the denominator: \(3x^2 + 6 = 3(x^2 + 2)\). Rewrite the integrand as \(\frac{x + 1}{3(x^2 + 2)}\).

Step 2: Split the integrand into two separate terms: \(\frac{x}{3(x^2 + 2)} + \frac{1}{3(x^2 + 2)}\). This allows us to handle each term individually.

Step 3: For the first term \(\frac{x}{3(x^2 + 2)}\), notice that the numerator is the derivative of the denominator \(x^2 + 2\). Use the substitution method: let \(u = x^2 + 2\), then \(du = 2x dx\). Adjust for the constant and rewrite the integral.

Step 4: For the second term \(\frac{1}{3(x^2 + 2)}\), recognize it as a standard integral form. Use the formula \(\int \frac{1}{a^2 + x^2} dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right)\), where \(a = \sqrt{2}\). Apply this formula to evaluate the integral.

Step 5: Combine the results of both integrals and evaluate the definite integral by substituting the limits \(x = 0\) and \(x = \sqrt{2}\). Simplify the expression to obtain the final result.

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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 fraction decomposition. Understanding these methods is crucial for evaluating complex integrals, as they allow for simplification and easier computation.

Definite Integrals

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

Rational Functions

Rational functions are ratios of polynomials, expressed as f(x) = P(x)/Q(x), where P and Q are polynomials. When integrating rational functions, techniques such as polynomial long division and partial fraction decomposition are often employed to simplify the integrand, making it easier to integrate.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.18. ∫ (from 0 to √2) (x + 1)/(3x² + 6) dx