Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)
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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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.1.40
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.
∫ (√x / (1 + x³)) dx
Hint: Let u = x^(3/2).
Verified step by step guidance1
Start by examining the integral: \(\int \frac{\sqrt{x}}{1 + x^{3}} \, dx\). Notice the hint suggests using the substitution \(u = x^{3/2}\), which is equivalent to \(u = x^{\frac{3}{2}}\).
Differentiate \(u\) with respect to \(x\) to find \(du\): Since \(u = x^{3/2}\), then \(\frac{du}{dx} = \frac{3}{2} x^{1/2}\). This implies \(du = \frac{3}{2} x^{1/2} dx\).
Rewrite the integral in terms of \(u\) and \(du\). From the expression for \(du\), solve for \(x^{1/2} dx\): \(x^{1/2} dx = \frac{2}{3} du\). Also, express the denominator \(1 + x^{3}\) in terms of \(u\): since \(u = x^{3/2}\), then \(u^{2} = x^{3}\), so \(1 + x^{3} = 1 + u^{2}\).
Substitute these expressions back into the integral: replace \(\sqrt{x} dx\) with \(\frac{2}{3} du\) and \(1 + x^{3}\) with \(1 + u^{2}\). The integral becomes \(\int \frac{\sqrt{x}}{1 + x^{3}} dx = \int \frac{\frac{2}{3} du}{1 + u^{2}}\).
Now, the integral simplifies to \(\frac{2}{3} \int \frac{1}{1 + u^{2}} du\), which is a standard integral. Recognize that \(\int \frac{1}{1 + u^{2}} du = \arctan(u) + C\). After integrating, substitute back \(u = x^{3/2}\) to express the answer in terms of \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. By letting u equal a function of x, we rewrite the integral in terms of u and du, making it easier to integrate. This technique is especially useful when the integral contains composite functions.
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07:33
Euler's Method
Algebraic Manipulation of Expressions
Algebraic manipulation involves rewriting expressions to reveal simpler forms or to match substitution patterns. In this problem, expressing √x and x³ in terms of powers of x helps identify a suitable substitution. Recognizing and manipulating exponents is key to simplifying the integral before applying substitution.
Recommended video:
6:36Simplifying Trig Expressions
Integration of Rational Functions
Integrals involving rational functions, where one polynomial is divided by another, often require substitution or partial fraction decomposition. Understanding how to handle integrals with expressions like 1/(1 + u) after substitution is essential. This knowledge helps in evaluating the integral once the substitution is applied.
Recommended video:
6:04Intro to Rational Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ arcsin(y) dy
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)
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Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ csc³(√θ) / √θ dθ
Textbook Question
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π
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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.
∫ √(9 - w²) dw / w²