Use Table 5.6 to evaluate the following definite integrals.
(c) ∫₃√₂^⁶ d𝓍/(𝓍² ―9)

검증된 단계별 안내

Step 1: Recognize that the integral involves a rational function of the form 1/(x² - a²). This suggests that the integral can be evaluated using a standard formula from Table 5.6: ∫ dx/(x² - a²) = (1/(2a)) ln| (x - a)/(x + a) | + C.

Step 2: Identify the value of 'a' in the denominator x² - 9. Here, x² - 9 can be rewritten as x² - 3², so a = 3.

Step 3: Apply the formula for the definite integral. Since the integral is definite, the limits of integration are given as √2 (lower limit) and 6 (upper limit). The formula becomes: ∫₃√₂⁶ dx/(x² - 9) = (1/(2 * 3)) [ln| (x - 3)/(x + 3) |] evaluated from x = √2 to x = 6.

Step 4: Simplify the constant factor outside the logarithm. The constant factor (1/(2 * 3)) simplifies to 1/6. So the integral becomes: (1/6) [ln| (x - 3)/(x + 3) |] evaluated from x = √2 to x = 6.

Step 5: Substitute the limits of integration into the logarithmic expression. First, substitute x = 6 into ln| (x - 3)/(x + 3) |, then substitute x = √2 into the same expression. Finally, subtract the result at x = √2 from the result at x = 6 to compute the definite integral.

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주요 개념

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Definite Integrals

A definite integral represents the signed area under a curve between two specified limits on the x-axis. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be solvable by basic antiderivatives. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these techniques is essential for simplifying complex integrals into manageable forms, allowing for easier evaluation.

Table of Integrals

A table of integrals is a reference tool that lists common integrals and their corresponding antiderivatives. It provides quick access to solutions for frequently encountered functions, saving time and effort in calculations. Familiarity with such tables can significantly aid in evaluating definite integrals efficiently, especially when dealing with standard forms.

Use Table 5.6 to evaluate the following definite integrals. (c) ∫₃√₂^⁶ d𝓍/(𝓍² ―9)