Textbook Question
Cosine limits Let n be a positive integer. Evaluate the following limits.
lim_x→0 (1 - cos xⁿ) / x²ⁿ
211
views
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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.91
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (2x +1)² dx
Verified step by step guidance1
Step 1: Recognize that the integral involves a polynomial expression squared, (2x + 1)². Expand this expression using the formula (a + b)² = a² + 2ab + b².
Step 2: Expand (2x + 1)² to get 4x² + 4x + 1. Rewrite the integral as ∫ (4x² + 4x + 1) dx.
Step 3: Break the integral into separate terms: ∫ 4x² dx + ∫ 4x dx + ∫ 1 dx. This allows you to integrate each term individually.
Step 4: Apply the power rule for integration to each term. For ∫ xⁿ dx, the result is (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration. Use this rule to integrate 4x², 4x, and 1.
Step 5: Combine the results of the individual integrals and include the constant of integration, C, to express the final indefinite integral.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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 denoted by the integral sign followed by the function and 'dx', indicating integration with respect to x. The result includes a constant of integration (C) since the derivative of a constant is zero, meaning multiple functions can yield the same derivative.
Recommended video:
05:04
Introduction to Indefinite Integrals
Polynomial Expansion
Polynomial expansion involves rewriting a polynomial expression in a simplified form, often using the binomial theorem. For example, expanding (2x + 1)² results in 4x² + 4x + 1. This step is crucial for integrating polynomials, as it allows for easier application of integration rules.
Recommended video:
07:00Taylor Polynomials
Power Rule of Integration
The power rule of integration states that the integral of x raised to the power n is (x^(n+1))/(n+1) + C, where n ≠ -1. This rule simplifies the process of finding indefinite integrals of polynomial functions, making it essential for solving integrals like ∫(2x + 1)² dx after expansion.
Recommended video:
04:04Power Rule for Indefinite Integrals
Related Practice
Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (x² / (x⁴ + x²)) dx
53
views
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) = 4cos (π (x-1)) on [0, 2]
253
views
Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (⁴√x³ + √x⁵) dx
88
views
Textbook Question
Cosine limits Let n be a positive integer. Evaluate the following limits.
lim_x→0 (1 - cosⁿ x) / x²
250
views
Textbook Question
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x⁴ - 50x² on [-1, 5]
214
views