Textbook Question
Checking Antiderivative Formulas
Verify the formulas in Exercises 57–62 by differentiation.
∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C
1
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.7.45
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(sin2x − csc²x)dx
Verified step by step guidance1
Rewrite the integral by separating it into two simpler integrals: \(\int (\sin 2x - \csc^2 x) \, dx = \int \sin 2x \, dx - \int \csc^2 x \, dx\).
Recall the integral formulas for each term: the integral of \(\sin 2x\) and the integral of \(\csc^2 x\). Specifically, use the substitution method for \(\sin 2x\) and the known antiderivative for \(\csc^2 x\).
For \(\int \sin 2x \, dx\), let \(u = 2x\), so \(du = 2 \, dx\) or \(dx = \frac{du}{2}\). Rewrite the integral in terms of \(u\) and integrate.
For \(\int \csc^2 x \, dx\), recall that the derivative of \(\cot x\) is \(-\csc^2 x\), so the integral of \(\csc^2 x\) is \(-\cot x\) plus a constant.
Combine the results from both integrals and add the constant of integration \(C\) to write the most general antiderivative. Finally, verify your answer by differentiating it to check if you get the original integrand.
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 and Antiderivative
An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. For example, ∫f(x)dx = F(x) + C, where F'(x) = f(x).
Recommended video:
05:04
Introduction to Indefinite Integrals
Integration of Trigonometric Functions
Integrating trigonometric functions like sin(2x) and csc²(x) requires knowledge of standard integral formulas and substitution techniques. For instance, ∫sin(2x)dx can be solved using a substitution for the inner function, while ∫csc²(x)dx is a standard integral equal to -cot(x) + C.
Recommended video:
6:04Introduction to Trigonometric Functions
Verification by Differentiation
After finding an antiderivative, verifying the solution by differentiating it ensures correctness. Differentiation of the proposed integral should return the original integrand. This step confirms the integral was computed accurately and helps identify any errors.
Recommended video:
05:53Finding Differentials
Related Practice
Textbook Question
Business and Economics
62. Production level Suppose that c(x)=x^3-20x^2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.
Textbook Question
A cone is formed from a circular piece of material of radius 1 meter by removing a section of angle θ and then joining the two straight edges. Determine the largest possible volume for the cone.
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫2x(1 − x⁻³) dx
Textbook Question
Finding Extrema from Graphs
In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
f(x) = |x|, −1 < x < 2
Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
h(x) = ³√x, −1 ≤ x ≤ 8