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Magazine Chapter 6: Problem 71
Evaluate each integral. $$ \int \frac{1}{x \sqrt{4 x^{2}-9}} d x $$
Short Answer
Expert verified
The solution to the integral is \( \frac{1}{3} \sec^{-1}\left(\frac{2x}{3}\right) + C \).
Step by step solution
01
Identify the Integral Type
First, notice that the integral involves a square root expression, which suggests that trigonometric substitution may be appropriate. The expression under the square root is \( 4x^2 - 9 \), which resembles the form \( a^2 - x^2 \) where \( a^2 = 9 \) and \( x^2 = 4x^2 \).
02
Make a Trigonometric Substitution
To simplify the integral, use the trigonometric substitution \( x = \frac{3}{2} \sec(\theta) \). This implies \( dx = \frac{3}{2} \sec(\theta) \tan(\theta) \,d\theta \). Substitute these into the integral.
03
Simplify the Integral
The expression under the square root becomes: \( \sqrt{4\left(\frac{3}{2}\sec(\theta)\right)^2 - 9} = \sqrt{9\sec^2(\theta) - 9} = \sqrt{9(\sec^2(\theta) - 1)} = 3\tan(\theta) \). So the integral now is \( \int \frac{1}{\frac{3}{2}\sec(\theta) \cdot 3\tan(\theta)} \cdot \frac{3}{2} \sec(\theta) \tan(\theta) \,d\theta \).
04
Evaluate the Integral
The \( \sec(\theta) \tan(\theta) \) terms cancel out, simplifying the integral to \( \int \frac{1}{3} \,d\theta \). This evaluates to \( \frac{1}{3} \theta + C \), where \( C \) is the constant of integration.
05
Back Substitute for \( x \)
Recall the substitution \( x = \frac{3}{2} \sec(\theta) \). Thus, \( \theta = \sec^{-1}\left(\frac{2x}{3}\right) \). Substitute back to get the result in terms of \( x \): \( \frac{1}{3} \sec^{-1}\left(\frac{2x}{3}\right) + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Substitution
Integrals involving square root expressions, especially those resembling forms like \( a^2 - x^2 \), often call for trigonometric substitution. This technique turns a complicated algebraic integral into a simpler trigonometric integral for easier evaluation. In our case, the expression \( \sqrt{4x^2 - 9} \) fits the pattern, suggesting we use secant function substitution.
- Set \( x = \frac{3}{2} \sec(\theta) \). This choice aligns with our goal to replace the square root with a trigonometric term, simplifying the integral structure.
- By differentiating, find \( dx = \frac{3}{2} \sec(\theta) \tan(\theta) \, d\theta \).
- Substitute these expressions into the integral, transforming it into a function of \( \theta \), thus avoiding square roots and maintaining trigonometric simplicity.
Definite Integral
While indefinite integrals have unlimited boundaries, definite integrals evaluate the total area under a curve within specific bounds. When solving such an integral, the limit of integration turns the potential constant \( C \) into a precise value, giving mathematical depth and utility.
- Limit of integration: In definite integrals, bounds confine calculations, making the area tangible and quantifiable.
- Evaluation: Compute \( \int \limits_{a}^{b} f(x) \,dx \) by substituting the upper and lower bounds into the antiderivative, then subtracting the results \( F(b) - F(a) \).
- Applications: Beyond calculating area, definite integrals help with solving volumes, average values, and other accumulated quantities of functions over intervals.
Indefinite Integral
The indefinite integral represents a family of functions, capturing the antiderivative of a given function without specific bounds. It offers generalized solutions by including a constant \( C \) due to its fundamental nature.
- Function Flexibility: An indefinite integral, \( \int f(x) \,dx = F(x) + C \), portrays a general antiderivative, reflecting all potential solutions by incorporating the constant.
- Key Elements: \( F(x) \) is the antiderivative, with \( C \) indicating that any function differing by this constant remains a valid solution.
- Problem Solving: By integrating functions and understanding behavior, anticipate broader applications in physics, engineering, and economics, where precise function models yield insights.
