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 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 8.6.52
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ csc³(√θ) / √θ dθ
Verified step by step guidance1
Start by making the substitution to simplify the integral. Let \( u = \sqrt{\theta} \), which implies \( \theta = u^2 \). Then, differentiate both sides to find \( d\theta \): \( d\theta = 2u \, du \).
Rewrite the integral in terms of \( u \). Since \( \sqrt{\theta} = u \), the integral becomes \( \int \frac{\csc^3(u)}{u} d\theta = \int \frac{\csc^3(u)}{u} \cdot 2u \, du = 2 \int \csc^3(u) \, du \). Notice how the \( u \) terms cancel out.
Focus now on evaluating \( \int \csc^3(u) \, du \). This is a standard integral that can be solved using a reduction formula or by expressing \( \csc^3(u) \) as \( \csc(u) \cdot \csc^2(u) \) and using integration by parts.
Recall the reduction formula for \( \int \csc^n(u) \, du \) when \( n \) is an odd integer: \[ \int \csc^n(u) \, du = -\frac{\csc^{n-2}(u) \cot(u)}{n-1} + \frac{n-2}{n-1} \int \csc^{n-2}(u) \, du \]. Apply this formula with \( n=3 \) to reduce the integral to a simpler form.
After applying the reduction formula, integrate the resulting simpler integral(s) and then substitute back \( u = \sqrt{\theta} \) to express the answer in terms of \( \theta \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Substitution
Trigonometric substitution involves replacing a variable or expression with a trigonometric function to simplify integrals, especially those involving roots or powers of trigonometric functions. It helps transform complicated integrals into more manageable forms by leveraging trigonometric identities.
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Introduction to Trigonometric Functions
Reduction Formulas
Reduction formulas are recursive relationships that express an integral with a higher power in terms of an integral with a lower power. They simplify the evaluation of integrals involving powers of trigonometric functions by breaking them down step-by-step.
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Guided course
5:59Recursive Formulas
Integration of Powers of Cosecant
Integrating powers of cosecant functions often requires using identities and reduction formulas due to their complexity. Understanding how to manipulate expressions like csc³(x) and apply appropriate substitutions is essential for solving such integrals.
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05:42Example 6: Integral of Secant & Cosecant
Related Practice
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 = x²/4, 0 ≤ x ≤ 2
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Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ arcsin(y) dy
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
Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₀^(π/6) √(1 + sin(x)) dx
(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))