Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
133. ∫ (sin²x) / (1 + sin²x) 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.PE.12
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(x + 1) / (x² (x − 1))] dx
Guida verificata passo dopo passo1
Start by examining the integrand \( \frac{x + 1}{x^{2}(x - 1)} \). Notice that the denominator is a product of powers of \( x \) and \( (x - 1) \), suggesting that partial fraction decomposition could be a useful method.
Set up the partial fraction decomposition for \( \frac{x + 1}{x^{2}(x - 1)} \) as follows: \(\n\[\n\)\( \frac{x + 1}{x^{2}(x - 1)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x - 1} \), \(\n\]\nwhere\) \( A \), \( B \), and \( C \) are constants to be determined.
Multiply both sides of the equation by the common denominator \( x^{2}(x - 1) \) to clear the fractions: \(\n\)\(\n\)\( x + 1 = A x (x - 1) + B (x - 1) + C x^{2} \).
Expand the right-hand side and collect like terms in powers of \( x \). Then, equate the coefficients of corresponding powers of \( x \) on both sides to form a system of equations for \( A \), \( B \), and \( C \).
Solve the system of equations to find the values of \( A \), \( B \), and \( C \). Once found, rewrite the integral as the sum of simpler integrals: \(\n\)\(\n\)\( \int \frac{A}{x} dx + \int \frac{B}{x^{2}} dx + \int \frac{C}{x - 1} dx \). Each of these can be integrated using basic integral formulas.
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Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function or a complicated expression.
Video consigliato:
04:27
Substitution With an Extra Variable
Partial Fraction Decomposition
Partial fraction decomposition breaks down a complex rational function into simpler fractions that are easier to integrate. This method is essential when integrating rational functions where the degree of the numerator is less than the degree of the denominator. It allows the integral to be expressed as a sum of simpler terms.
Video consigliato:
Percorso guidato
10:07Partial Fraction Decomposition: Distinct Linear Factors
Properties of Rational Functions
Understanding rational functions, which are ratios of polynomials, is crucial for integration. Recognizing factors in the denominator and numerator helps in choosing the right method, such as substitution or partial fractions. Analyzing the function's structure guides the simplification and integration process.
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Percorso guidato
06:21Properties of Functions
Pratica correlata
Domanda del libro di testo
A brief calculation shows that if 0 ≤ x ≤ 1, then the second derivative of
f(x) = √(1 + x⁴)
lies between 0 and 8.
Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1
with an error no greater than 10⁻³ in absolute value using the Trapezoidal Rule?
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Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
117. ∫ dr / (1 + √r)
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Domanda del libro di testo
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ arccos(x / 2) dx
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Domanda del libro di testo
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x² ln(x) dx
31
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Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ x dx / √(2 − x)
23
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