Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx
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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 8.5.48
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x√x + 9) dx
Verified step by step guidance1
Rewrite the integral to clarify the expression inside the square root: the integral is \(\int \frac{1}{x \sqrt{x + 9}} \, dx\).
Use a substitution to simplify the square root term. Let \(u = x + 9\), so that \(du = dx\) and \(x = u - 9\).
Rewrite the integral in terms of \(u\): replace \(x\) with \(u - 9\) and \(dx\) with \(du\), so the integral becomes \(\int \frac{1}{(u - 9) \sqrt{u}} \, du\).
Express the integrand as \(\frac{1}{(u - 9) u^{1/2}} = \frac{1}{u^{1/2} (u - 9)}\) and consider using partial fraction decomposition or another substitution to simplify this expression further.
Set up the partial fraction decomposition or an appropriate substitution to break down the integrand into simpler terms that can be integrated individually.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration Techniques
Integration techniques are methods used to find antiderivatives or evaluate integrals. Common techniques include substitution, integration by parts, and partial fractions. Choosing the right technique depends on the form of the integrand, such as recognizing when a substitution simplifies the integral.
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Integration by Parts for Definite Integrals
Substitution Method
The substitution method involves changing variables to simplify an integral. By letting a new variable equal a function inside the integral, the integral can often be rewritten in a simpler form. This is especially useful when the integrand contains composite functions or expressions under roots.
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07:33Euler's Method
Handling Radicals in Integrals
Integrals involving radicals, such as square roots, often require algebraic manipulation or substitution to simplify the expression. Recognizing how to rewrite radicals and expressions like √(x + 9) can help transform the integral into a more manageable form for evaluation.
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06:13Limits of Rational Functions with Radicals
Related Practice
Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ
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 1 of (-x ln|x| dx)
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Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ + 4)) dx
Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ 8 cot⁴(t) dt