Domanda del libro di testo
37–56. Integrals Evaluate each integral.
∫₀⁴ sech²√x / √x dx
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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
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
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.47
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.47Capitolo 7, Problema 7.3.47
37–56. Integrals Evaluate each integral.
∫ dx/(8 – x²), x > 2√2
Guida verificata passo dopo passo1
Recognize that the integral is of the form \(\int \frac{dx}{a^2 - x^2}\), where \(a^2 = 8\), so \(a = 2\sqrt{2}\).
Recall the standard integral formula: \(\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a + x}{a - x} \right| + C\), valid for \(|x| > a\).
Since the problem states \(x > 2\sqrt{2}\), the condition for the formula applies directly.
Substitute \(a = 2\sqrt{2}\) into the formula to write the integral in terms of \(x\) and \(a\).
Write the final expression for the integral as \(\frac{1}{2 \cdot 2\sqrt{2}} \ln \left| \frac{2\sqrt{2} + x}{2\sqrt{2} - x} \right| + C\), simplifying the coefficient if desired.
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Integration of Rational Functions
This involves integrating functions expressed as ratios of polynomials. Recognizing the form of the integrand helps determine the appropriate method, such as partial fractions or substitution, to simplify and evaluate the integral.
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Intro to Rational Functions
Inverse Hyperbolic Functions
Integrals involving expressions like 1/(a² - x²) often lead to inverse hyperbolic functions such as arctanh or arcsinh. Understanding their definitions and derivatives is essential for correctly evaluating these integrals.
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4:49Inverse Cosine
Domain Restrictions and Absolute Values
The condition x > 2√2 restricts the domain, affecting the sign of expressions under square roots or logarithms. Properly handling these restrictions ensures the correct form of the antiderivative and avoids extraneous solutions.
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Percorso guidato
5:10Finding the Domain and Range of a Graph
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