Domanda del libro di testo
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = xe^x-e^x
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 7, Problema 7.6.111
Verify the integration formulas in Exercises 111–114.
111. ∫ (arctan x) / x² dx = ln x - 1/2 ln(1 + x²) - arctan x / x + C
Guida verificata passo dopo passo1
Identify the integral to verify: \(\int \frac{\arctan x}{x^{2}} \, dx\).
Consider using integration by parts. Let \(u = \arctan x\) and \(dv = \frac{1}{x^{2}} dx\).
Compute the derivatives and integrals needed: \(du = \frac{1}{1 + x^{2}} dx\) and \(v = -\frac{1}{x}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), which gives \(-\frac{\arctan x}{x} - \int \left(-\frac{1}{x} \cdot \frac{1}{1 + x^{2}}\right) dx\).
Simplify the remaining integral \(\int \frac{1}{x(1 + x^{2})} dx\) by using partial fraction decomposition or substitution, then combine all parts to match the given formula.
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Integration by Parts
Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du. Choosing appropriate u and dv is crucial to simplify the integral effectively.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Derivative and Integral of Inverse Trigonometric Functions
Understanding the derivatives and integrals of inverse trigonometric functions like arctan(x) is essential. For example, d/dx [arctan x] = 1/(1 + x²). This knowledge helps in manipulating integrals involving arctan(x) and recognizing patterns during integration.
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06:35Derivatives of Other Inverse Trigonometric Functions
Logarithmic Integration and Simplification
Integrals often result in logarithmic expressions, especially when integrating rational functions. Recognizing when to rewrite expressions using logarithm properties, such as ln(a) - ln(b) = ln(a/b), aids in simplifying the final answer and verifying given formulas.
Video consigliato:
7:30Logarithms Introduction
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