Domanda del libro di testo
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋₂¹ (1 / x⁴) 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 8.5.50
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ + 4)) dx
Guida verificata passo dopo passo1
Start by rewriting the integral to clearly see the expression: \(\int \frac{1}{x^{6}(x^{5} + 4)} \, dx\).
Consider a substitution to simplify the integral. Notice that \(x^{5}\) appears inside the parentheses, so let \(u = x^{5} + 4\).
Differentiate \(u\) with respect to \(x\) to find \(du\): \(du = 5x^{4} \, dx\), which implies \(dx = \frac{du}{5x^{4}}\).
Rewrite the integral in terms of \(u\) and \(x\): substitute \(x^{5} + 4\) with \(u\) and \(dx\) with \(\frac{du}{5x^{4}}\). Also, express the remaining powers of \(x\) in terms of \(u\) if possible.
Simplify the integral after substitution and look for further algebraic manipulation or partial fraction decomposition to integrate with respect to \(u\).
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Integration of Rational Functions
This involves integrating functions expressed as the ratio of two polynomials. Techniques often include algebraic manipulation, partial fraction decomposition, or substitution to simplify the integral into manageable parts.
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6:04
Intro to Rational Functions
Partial Fraction Decomposition
A method used to break down complex rational expressions into simpler fractions that are easier to integrate. It requires factoring the denominator and expressing the integrand as a sum of simpler rational terms.
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Percorso guidato
10:07Partial Fraction Decomposition: Distinct Linear Factors
Substitution Method
A technique where a part of the integrand is replaced with a new variable to simplify the integral. It is especially useful when the integral contains a composite function or when the derivative of a function appears within the integrand.
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07:33Euler's Method
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