Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (x² - 2x + 1) e^(2x) dx
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 8.8.4
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀⁴ dx / √(4 − x)
Verified step by step guidance1
Identify the integral to be solved: \(\int_0^4 \frac{dx}{\sqrt{4 - x}}\).
Recognize that the integrand involves a square root of a linear function, suggesting a substitution to simplify the integral.
Use the substitution \(u = 4 - x\), which implies \(du = -dx\) or \(dx = -du\). Also, change the limits of integration accordingly: when \(x=0\), \(u=4\); when \(x=4\), \(u=0\).
Rewrite the integral in terms of \(u\): \(\int_{u=4}^{u=0} \frac{-du}{\sqrt{u}} = \int_0^4 \frac{du}{\sqrt{u}}\) after reversing the limits to keep the integral positive.
Evaluate the integral \(\int_0^4 u^{-1/2} du\) by applying the power rule for integration: \(\int u^n du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\).
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.
Improper Integrals and Convergence
An integral is improper if the interval of integration is infinite or the integrand becomes unbounded within the interval. Understanding convergence means determining whether the integral approaches a finite value. In this problem, the integrand has a singularity at x = 4, so recognizing the integral as improper is essential.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Substitution Method for Integration
The substitution method simplifies integrals by changing variables to transform the integrand into a more manageable form. For example, setting a new variable equal to the expression inside the square root can help rewrite the integral in terms of a simpler function, making it easier to evaluate.
Recommended video:
07:33Euler's Method
Evaluating Definite Integrals with Square Root Functions
Integrals involving square roots often require algebraic manipulation or trigonometric substitution to evaluate. Recognizing the form √(a - x) allows the use of substitution or standard integral formulas to find antiderivatives and then apply the limits to compute the definite integral.
Recommended video:
05:43Definition of the Definite Integral
Related Practice
Textbook Question
In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.
∫ 2y⁴ / (y³ - y² + y - 1) dy
1
views
Textbook Question
Solve the initial value problems in Exercises 67–70 for x as a function of t.
(t + 1) (dx/dt) = x² + 1 (for t > -1), x(0) = 0
Textbook Question
Volume of water in a swimming pool
A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth h(x) of the water at 5-ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with n = 10 applied to the integral
V = ∫ from 0 to 50 of 30 · h(x) dx.
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² tan⁻¹(x / 2) dx
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.
∫ ((2ˣ - 1) / 3ˣ) dx