Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
1
views
Ch. 8 - Techniques of Integration
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 8, Problem 8.2.22
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ e^(-y) cos(y) dy
Verified step by step guidance1
Identify the integral to solve: \(\int e^{-y} \cos(y) \, dy\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u\) and \(dv\) wisely. For this integral, let \(u = \cos(y)\) and \(dv = e^{-y} dy\).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = -\sin(y) dy\), and integrate \(dv\) to get \(v = \int e^{-y} dy = -e^{-y}\).
Apply the integration by parts formula: substitute \(u\), \(v\), \(du\) into \(\int u \, dv = uv - \int v \, du\) and simplify the resulting integral.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the integral effectively.
Recommended video:
06:18
Integration by Parts for Definite Integrals
Integration of Exponential Functions
Exponential functions like e^(-y) have straightforward antiderivatives, often involving a constant factor adjustment. Recognizing how to integrate e^(-y) is essential, as it frequently appears in integrals combined with trigonometric functions.
Recommended video:
05:11Integrals of General Exponential Functions
Integration of Trigonometric Functions
Integrals involving trigonometric functions such as cos(y) often require knowledge of their antiderivatives, like sin(y) for cos(y). When combined with other functions, these integrals may need repeated application of integration techniques.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Solve the initial value problems in Exercises 53–56 for y as a function of x.
√(x² - 9) (dy/dx) = 1, where x > 3, y(5) = ln 3
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (sec t + cot t)² dt
Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁴ [x / (x² + 9)^(2/5)] dx
1
views
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 / √(x^2 - 4x + 5) dx
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ dx / (8 + 2x²) from 0 to 2