Domanda del libro di testo
Evaluate the integrals in Exercises 1–22.
∫ sin⁴(2x) cos(2x) dx
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Ch. 8 - Techniques of Integration
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 8, Problema 8.6.20
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ tan^(-1)(x) / x² dx
Guida verificata passo dopo passo1
Recognize that the integral is \( \int \frac{\tan^{-1}(x)}{x^2} \, dx \), where \( \tan^{-1}(x) \) is the inverse tangent function, also written as \( \arctan(x) \).
Consider using integration by parts, since the integrand is a product of functions: one involving \( \arctan(x) \) and the other involving \( \frac{1}{x^2} \). Set \( u = \arctan(x) \) and \( dv = \frac{1}{x^2} dx \).
Compute \( du \) and \( v \):
- \( du = \frac{1}{1+x^2} dx \) because the derivative of \( \arctan(x) \) is \( \frac{1}{1+x^2} \).
- \( v = \int x^{-2} dx = -x^{-1} = -\frac{1}{x} \).
Apply the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Substitute the expressions for \( u, v, du \) to rewrite the integral.
Simplify the resulting integral and evaluate it using the table of integrals if necessary, focusing on the integral \( \int \frac{1}{x(1+x^2)} dx \) that appears after substitution.
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Inverse Trigonometric Functions
Inverse trigonometric functions, like arctan(x), are the inverses of the standard trig functions and return angles given a ratio. Understanding their properties and derivatives is essential for integrating expressions involving these functions.
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Derivatives of Other Inverse Trigonometric Functions
Integration Techniques
Integration techniques such as integration by parts or substitution are often required to solve integrals involving products or compositions of functions, especially when standard formulas are not directly applicable.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
Use of Integral Tables
Integral tables provide formulas for common integrals, including those involving inverse trig functions. Knowing how to locate and apply these formulas can simplify the evaluation of complex integrals.
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Percorso guidato
08:01Integration Using Partial Fractions
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