Textbook Question
Solving initial value problems Find the solution of the following initial value problems.
g'(x) = 7x(x⁶ - 1/7); g(1) = 2
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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 4.9.65
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((1 + √x)/x)dx
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Rewrite the integrand to simplify the expression. Split the fraction into two terms: \( \frac{1}{x} + \frac{\sqrt{x}}{x} \). Simplify further to get \( \frac{1}{x} + x^{-1/2} \).
Express the integrand in terms of powers of \(x\): \( x^{-1} + x^{-1/2} \). This makes it easier to apply the power rule for integration.
Apply the power rule for integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \(n \neq -1\). For \(x^{-1}\), recall that \( \int x^{-1} dx = \ln|x| + C \).
Integrate each term separately: \( \int x^{-1} dx = \ln|x| \) and \( \int x^{-1/2} dx = \frac{x^{1/2}}{1/2} = 2x^{1/2} \).
Combine the results to write the final integral: \( \ln|x| + 2x^{1/2} + C \), where \(C\) is the constant of integration. Check your work by differentiating this result to ensure it matches the original integrand.
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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 antidifferentiation, where we seek a function F(x) such that F'(x) equals the integrand.
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Integration Techniques
To solve integrals, various techniques can be employed, such as substitution, integration by parts, or partial fraction decomposition. In this case, simplifying the integrand, which is a rational function, can make the integration process more manageable. Recognizing patterns and applying appropriate techniques is crucial for finding the correct antiderivative.
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Verification by Differentiation
After finding an indefinite integral, it is essential to verify the result by differentiating the antiderivative obtained. This step ensures that the differentiation of the antiderivative returns the original integrand. This verification process is a fundamental practice in calculus, confirming the correctness of the integration performed.
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Related Practice
Textbook Question
Explain how to apply the First Derivative Test.
Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x⁴ + 4x³
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Textbook Question
{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = cos x
Textbook Question
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = sin² x
Textbook Question
Suppose the position of an object moving horizontally after seconds is given by the function s(t) = 32t - t⁴, where 0 ≤ t ≤ 3 and s is measured in feet, with s > 0 corresponding to positions to the right of the origin. When is the object farthest to the right?