Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x - x²) / x dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 8.2.38
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x⁵ e³ˣ dx
Verified step by step guidance1
Identify the integral to solve: \(\int x^{5} e^{3x} \, dx\).
Recognize that this integral involves a product of a polynomial \(x^{5}\) and an exponential function \(e^{3x}\), which suggests using integration by parts.
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Choose \(u = x^{5}\) (since its derivative simplifies) and \(dv = e^{3x} \, dx\) (since it integrates easily).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = 5x^{4} \, dx\), and integrate \(dv\) to get \(v = \frac{1}{3} e^{3x}\).
Apply the integration by parts formula: \(\int x^{5} e^{3x} \, dx = x^{5} \cdot \frac{1}{3} e^{3x} - \int \frac{1}{3} e^{3x} \cdot 5x^{4} \, dx\). This reduces the original integral to a similar integral with a lower power of \(x\), which you can solve by repeating integration by parts until the polynomial term is eliminated.
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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. This method is especially useful when integrating products like x⁵ and e³ˣ.
Recommended video:
06:18
Integration by Parts for Definite Integrals
Choosing u and dv
Selecting which part of the integrand to assign as u and which as dv is crucial for simplifying the integral. Typically, u is chosen as a polynomial (like x⁵) because its derivative simplifies, and dv as an exponential or trigonometric function, which is easy to integrate.
Recommended video:
07:51Choosing a Convergence Test
Repeated Application of Integration by Parts
When the polynomial degree is high, integration by parts may need to be applied multiple times, reducing the power of x step-by-step. This iterative process continues until the polynomial term is eliminated, allowing the integral to be evaluated fully.
Recommended video:
08:23Repeated Integration by Parts
Related Practice
Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 1 to ∞ of ((1 / (e^x - 2^x)) dx)
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Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x dx) / (25 + 4x²)
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Textbook Question
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ ln(x + x²) dx
Textbook Question
Use any method to evaluate the integrals in Exercises 65–70.
∫ sin³(x) / cos⁴(x) dx
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 √(2x - x^2) dx