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 8.4.50
Capitolo 8, Problema 8.4.50

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx

Guida verificata passo dopo passo
1
Start by completing the square for the expression inside the square root: \(x^{2} - 2x + 5\). To do this, rewrite it as \(\left(x^{2} - 2x + 1\right) + 4\) because \(1\) is the square of half the coefficient of \(x\).
Express the completed square form explicitly: \(\left(x - 1\right)^{2} + 4\). So the integral becomes \(\int \frac{1}{\sqrt{\left(x - 1\right)^{2} + 4}} \, dx\).
Recognize that the integral now has the form \(\int \frac{1}{\sqrt{u^{2} + a^{2}}} \, du\) where \(u = x - 1\) and \(a = 2\). This suggests using the trigonometric substitution \(u = a \tan \theta\), or \(x - 1 = 2 \tan \theta\).
Differentiate the substitution to find \(dx\): since \(x - 1 = 2 \tan \theta\), then \(dx = 2 \sec^{2} \theta \, d\theta\).
Rewrite the integral in terms of \(\theta\) using the substitution and simplify the square root using the identity \(1 + \tan^{2} \theta = \sec^{2} \theta\). Then proceed to integrate with respect to \(\theta\).

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:
12m

Concetti chiave

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

Completing the Square

Completing the square is a method used to rewrite a quadratic expression in the form ax² + bx + c as a perfect square plus or minus a constant. This technique simplifies the integrand, making it easier to identify the appropriate substitution, especially for integrals involving square roots of quadratic expressions.
Video consigliato:
05:22
Completing the Square to Rewrite the Integrand

Trigonometric Substitution

Trigonometric substitution is a technique used to evaluate integrals involving square roots of quadratic expressions by substituting variables with trigonometric functions. Depending on the form (x² ± a² or a² - x²), substitutions like x = a sec θ or x = a tan θ transform the integral into a trigonometric integral that is easier to solve.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Integration of Functions Involving Square Roots

Integrals containing square roots of quadratic expressions often require algebraic manipulation and substitution to simplify. Understanding how to handle these integrals involves recognizing patterns, applying completing the square, and using trigonometric identities to rewrite and integrate the function effectively.
Video consigliato:
07:01
Integrals Involving Natural Logs: Substitution
Pratica correlata
Domanda del libro di testo

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from e to e^e of (ln(ln x) dx)

36
views
Domanda del libro di testo

In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x² dx) / ((x - 1)(x² + 2x + 1))

31
views
Domanda del libro di testo

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 0 to 1 of (dt / (t - sin t))

(Hint: t ≥ sin t for t ≥ 0)

23
views
Domanda del libro di testo

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (2x + 1) / (x² - 7x + 12) dx

31
views
Domanda del libro di testo

Evaluate the integrals in Exercises 33–52.

∫ cot⁶(2x) dx

19
views
Domanda del libro di testo

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (x dx) / (25 + 4x²)

22
views