Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
48. ∫ √(9 - 4x²) dx
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.2.20
9–40. Integration by parts Evaluate the following integrals using integration by parts.
20. ∫ sin⁻¹(x) dx
Verified step by step guidance1
Identify the integral to solve: \(\int \sin^{-1}(x) \, dx\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u\) and \(dv\) wisely. Let \(u = \sin^{-1}(x)\) because its derivative simplifies, and let \(dv = dx\) because it is easy to integrate.
Compute \(du\) and \(v\):
- \(du = \frac{1}{\sqrt{1 - x^2}} \, dx\) (derivative of \(\sin^{-1}(x)\)),
- \(v = x\) (integral of \(dx\)).
Apply the integration by parts formula:
\(\int \sin^{-1}(x) \, dx = x \sin^{-1}(x) - \int x \cdot \frac{1}{\sqrt{1 - x^2}} \, dx\).
Next, focus on evaluating the remaining integral \(\int \frac{x}{\sqrt{1 - x^2}} \, dx\).
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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 derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the integral effectively.
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Integration by Parts for Definite Integrals
Inverse Trigonometric Functions
Inverse trigonometric functions, like sin⁻¹(x), are the inverses of trigonometric functions and have specific derivatives. For example, the derivative of sin⁻¹(x) is 1/√(1 - x²). Understanding these derivatives helps in setting up the parts for integration.
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06:35Derivatives of Other Inverse Trigonometric Functions
Differentiation and Integration of Algebraic and Transcendental Functions
Integrating functions involving inverse trigonometric terms often requires combining algebraic manipulation with knowledge of derivatives and integrals of transcendental functions. Recognizing how to differentiate and integrate these functions is essential for applying integration by parts successfully.
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Related Practice
Textbook Question
5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
9. 4/(x⁵ - 5x³ + 4x)
Textbook Question
7–64. Integration review Evaluate the following integrals.
20. ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) dx
Textbook Question
5. What is a reduction formula?
Textbook Question
50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:
53. ∫ lnⁿ(x) dx = x lnⁿ(x) - n ∫ lnⁿ⁻¹(x) dx
Textbook Question
River flow rates
The following figure shows the discharge rate r(t) of the Snoqualmie River near Carnation, Washington, starting on a February day when the air temperature was rising. The variable t is the number of hours after midnight, r(t) is given in millions of cubic feet per hour, and ∫(0 to 24) r(t) dt equals the total amount of water that flows by the town of Carnation over a 24-hour period. Estimate ∫(0 to 24) r(t) dt using the Trapezoidal Rule and Simpson's Rule with the following values of n.
n = 6