Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (2x +1)² dx
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.93
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ ((1/x²) - (2/(x⁵⸍²))) dx
Verified step by step guidance1
Rewrite the integrand to simplify the terms. The given integral is ∫ ((1/x²) - (2/(x^(5/2)))) dx. Express each term using exponents: ∫ (x^(-2) - 2x^(-5/2)) dx.
Apply the power rule for integration. Recall that the integral of x^n is (x^(n+1))/(n+1), provided n ≠ -1. Use this rule for each term in the integrand.
For the first term, x^(-2), integrate it using the power rule: ∫ x^(-2) dx = (x^(-2+1))/(-2+1) = x^(-1)/(-1) = -1/x.
For the second term, -2x^(-5/2), integrate it using the power rule: ∫ -2x^(-5/2) dx = -2 * (x^(-5/2+1))/(-5/2+1) = -2 * (x^(-3/2))/(-3/2). Simplify the coefficient.
Combine the results of the two integrals and add the constant of integration, C, to express the final indefinite integral: ∫ ((1/x²) - (2/(x^(5/2)))) dx = -1/x + (4/3)x^(-3/2) + C.
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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 of integration 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 F(x) such that F'(x) equals the integrand.
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Power Rule for Integration
The power rule for integration is a fundamental technique used to integrate functions of the form x^n, where n is any real number except -1. According to this rule, the integral of x^n is (x^(n+1))/(n+1) + C. This rule simplifies the process of integrating polynomial and rational functions, making it essential for solving many integral problems.
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Rational Functions
Rational functions are ratios of two polynomials, expressed as P(x)/Q(x), where P and Q are polynomials. When integrating rational functions, techniques such as polynomial long division or substitution may be necessary to simplify the integrand. Understanding how to manipulate and integrate these functions is crucial for solving integrals involving terms like 1/x² and 2/x⁵.
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Related Practice
Textbook Question
Optimal popcorn box A small popcorn box is created from a 12" x 12" sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box? <IMAGE>
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).
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.
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Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (12/x)dx
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)