{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,
∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C


Graph the following functions and find the area under the curve on the given interval.
f(x) = 1/(x√(x² - 36)), [12/√3 , 12]

Verified step by step guidance

1. First, rewrite the integral for the area under the curve as \(\int_{12/\sqrt{3}}^{12} \frac{1}{x \sqrt{x^2 - 36}} \, dx\).
2. Use the substitution \(x = 6 \sec u\) because \(\sqrt{x^2 - 36}\) suggests a secant substitution (since \(36 = 6^2\)). Then, compute \(dx\) in terms of \(du\).
3. Express the integral entirely in terms of \(u\) by substituting \(x = 6 \sec u\), \(dx = 6 \sec u \tan u \, du\), and \(\sqrt{x^2 - 36} = 6 \tan u\).
4. Simplify the integral after substitution to get an integral involving \(\sec^3 u\), which matches the form given in the reduction formula: \(\int \sec^3 u \, du\).
5. Apply the reduction formula \(\int \sec^3 u \, du = \frac{1}{2} (\sec u \tan u + \ln |\sec u + \tan u|) + C\) to evaluate the integral, then substitute back to \(x\) using \(u = \sec^{-1}(x/6)\) and evaluate the definite integral at the given limits.

Key Concepts

Integration Using Reduction Formulas

Reduction formulas simplify complex integrals by expressing them in terms of simpler integrals of the same type. For example, the integral of sec³u can be evaluated using a known reduction formula, which breaks it down into a combination of sec u tan u and a logarithmic term. This technique is essential for handling powers of trigonometric functions.

Definite Integrals and Area Under a Curve

A definite integral calculates the net area between a function's graph and the x-axis over a specified interval. Evaluating ∫ f(x) dx from a to b gives the total area under f(x) between x = a and x = b, accounting for regions above and below the axis. This concept connects integration to geometric interpretation.

Trigonometric Substitution in Integration

Trigonometric substitution is a method used to simplify integrals involving expressions like √(x² - a²) by substituting x with a trigonometric function (e.g., x = a sec θ). This transforms the integral into a trigonometric integral, which can be easier to evaluate using known identities and formulas, such as the integral of sec³u.