Skip to main content
Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.PE.10
Capitolo 8, Problema 8.PE.10

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [x / (x² + 4x + 3)] dx

Guida verificata passo dopo passo
1
Start by examining the integral: \(\int \frac{x}{x^{2} + 4x + 3} \, dx\). Notice that the denominator is a quadratic expression that can potentially be factored.
Factor the quadratic in the denominator: \(x^{2} + 4x + 3 = (x + 1)(x + 3)\). This will help in simplifying the integral or setting up partial fractions.
Since the degree of the numerator is less than the degree of the denominator, consider using partial fraction decomposition. Express \(\frac{x}{(x + 1)(x + 3)}\) as \(\frac{A}{x + 1} + \frac{B}{x + 3}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides by \((x + 1)(x + 3)\) to clear the denominators: \(x = A(x + 3) + B(x + 1)\). Expand and collect like terms to form an equation in terms of \(x\).
Equate the coefficients of like powers of \(x\) on both sides to solve for \(A\) and \(B\). Once \(A\) and \(B\) are found, rewrite the integral as the sum of two simpler integrals and integrate each term separately.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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 function and its derivative.
Video consigliato:
04:27
Substitution With an Extra Variable

Partial Fraction Decomposition

Partial fraction decomposition breaks down a rational function into simpler fractions that are easier to integrate. This method is applicable when the denominator can be factored into linear or quadratic terms. After decomposition, each simpler fraction can be integrated using basic integral formulas.
Video consigliato:
Percorso guidato
10:07
Partial Fraction Decomposition: Distinct Linear Factors

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. Factoring the denominator, such as x² + 4x + 3, helps in simplifying the integral and applying partial fraction decomposition. Recognizing factorable quadratics is essential for breaking down complex rational expressions.
Video consigliato:
07:00
Taylor Polynomials
Pratica correlata
Domanda del libro di testo

Heat capacity of a gas

Heat capacity 

C_v

is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).

The heat capacity of oxygen depends on its temperature T and satisfies the formula

C_v = 8.27 + 10^(-5) * (26T − 1.87T²)

Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for

20°C ≤ T ≤ 675°C.

20
views
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.

∫ sinx·cos²x dx

22
views
Domanda del libro di testo

135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:

(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.

22
views
Domanda del libro di testo

Evaluate the integrals in Exercises 33–36.

∫ [1 / √(9 - x²)] dx

18
views
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.

129. ∫ (x^(ln x) * ln x) / x dx

18
views
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.

∫ t dt / √(9 − 4t²)

13
views