Textbook Question
Evaluating integrals Evaluate the following integrals.
∫ 𝓍⁷ √(𝓍⁴ + 1d𝓍)
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
2:23 minutesProblem 4.9.57
Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (1/x√(36x² - 36))dx
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Rewrite the integral in a more manageable form. Factor out constants from the square root in the denominator: \( \int \frac{1}{x \sqrt{36x^2 - 36}} \, dx \). Factor out \(36\) from the square root: \( \sqrt{36x^2 - 36} = \sqrt{36(x^2 - 1)} = 6\sqrt{x^2 - 1} \). The integral becomes \( \int \frac{1}{6x\sqrt{x^2 - 1}} \, dx \).
Recognize the structure of the integrand. The term \( \sqrt{x^2 - 1} \) suggests a trigonometric substitution. Use the substitution \( x = \sec(\theta) \), which implies \( dx = \sec(\theta)\tan(\theta) \, d\theta \) and \( \sqrt{x^2 - 1} = \tan(\theta) \).
Substitute \( x = \sec(\theta) \) into the integral. The integral becomes \( \int \frac{1}{6\sec(\theta)\tan(\theta) \cdot \sec(\theta)} \sec(\theta)\tan(\theta) \, d\theta \). Simplify the expression: \( \int \frac{1}{6\sec^2(\theta)} \, d\theta = \frac{1}{6} \int \cos^2(\theta) \, d\theta \).
Simplify further using the trigonometric identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). The integral becomes \( \frac{1}{6} \int \frac{1 + \cos(2\theta)}{2} \, d\theta = \frac{1}{12} \int (1 + \cos(2\theta)) \, d\theta \). Split the integral: \( \frac{1}{12} \int 1 \, d\theta + \frac{1}{12} \int \cos(2\theta) \, d\theta \).
Evaluate each term. The first term is \( \frac{1}{12} \theta \). For the second term, use the integral of \( \cos(2\theta) \), which is \( \frac{1}{2}\sin(2\theta) \). Combine the results: \( \frac{1}{12} \theta + \frac{1}{24}\sin(2\theta) + C \). Finally, back-substitute \( \theta \) using \( x = \sec(\theta) \), where \( \theta = \sec^{-1}(x) \), to express the solution in terms of \( x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integrals
Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, where we seek a function whose derivative matches the given function.
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Substitution Method
The substitution method is a technique used in integration to simplify the integrand by changing variables. This involves identifying a part of the integrand that can be replaced with a new variable, making the integral easier to solve. After integration, the original variable is substituted back to express the result in terms of the initial variable.
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Differentiation Check
A differentiation check involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. This process ensures that the antiderivative found is accurate by confirming that differentiating it returns the original integrand. It serves as a crucial step in validating the integration process.
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