Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ ((1/x²) - (2/(x⁵⸍²))) dx
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.R.99
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (12/x)dx
Verified step by step guidance1
Step 1: Recognize that the integral ∫ (12/x) dx involves a basic logarithmic rule. Recall that the integral of 1/x with respect to x is ln|x|.
Step 2: Factor out the constant 12 from the integral. This simplifies the expression to 12 ∫ (1/x) dx.
Step 3: Apply the logarithmic integration rule. The integral of 1/x dx is ln|x|, so the expression becomes 12 ln|x|.
Step 4: Add the constant of integration, C, to account for the indefinite nature of the integral. The result is 12 ln|x| + C.
Step 5: Verify the result by differentiating 12 ln|x| + C. The derivative of ln|x| is 1/x, and multiplying by 12 gives back the original integrand, confirming the solution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integral
An indefinite integral represents a family of functions whose derivative is the integrand. It is expressed without specific limits and includes a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation.
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Introduction to Indefinite Integrals
Integration of Rational Functions
Rational functions are ratios of polynomials. To integrate a rational function, one often uses techniques such as substitution or partial fraction decomposition. In the case of the integral ∫ (12/x)dx, recognizing that this is a simple rational function allows for straightforward integration using the natural logarithm.
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6:04Intro to Rational Functions
Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately 2.71828. It is particularly important in calculus because the derivative of ln(x) is 1/x, making it a key function when integrating expressions involving 1/x, such as in the integral ∫ (12/x)dx.
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05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).
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Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 (3 sin² 2Θ) / Θ²
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Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫(1 + 3 cosΘ) dΘ
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Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 2Θ cot 3Θ
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Textbook Question
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 10x² / (x² + 3)
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