Domanda del libro di testo
23-64. Integration Evaluate the following integrals.
23. ∫ [3 / ((x - 1)(x + 2))] dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.2.20
9–40. Integration by parts Evaluate the following integrals using integration by parts.
20. ∫ sin⁻¹(x) dx
Guida verificata passo dopo passo1
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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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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06:18
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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Percorso guidato
04:22Integrals Resulting in Basic Trig Functions Example 1
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