Domanda del libro di testo
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 / √(x^2 - 4x + 5) dx
40
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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 8.1.52
Use the substitution u = tan x to evaluate the integral
∫ dx / (1 + sin² x).
Guida verificata passo dopo passo1
Start with the given integral: \(\int \frac{dx}{1 + \sin^{2} x}\).
Use the substitution \(u = \tan x\). Recall that \(\frac{du}{dx} = \sec^{2} x\), so \(dx = \frac{du}{\sec^{2} x}\).
Express \(\sin^{2} x\) and \(\sec^{2} x\) in terms of \(u\). Since \(u = \tan x\), we have \(\sin x = \frac{u}{\sqrt{1+u^{2}}}\) and \(\sec^{2} x = 1 + \tan^{2} x = 1 + u^{2}\).
Rewrite the integral in terms of \(u\): replace \(dx\) with \(\frac{du}{1+u^{2}}\) and \(\sin^{2} x\) with \(\frac{u^{2}}{1+u^{2}}\). Simplify the denominator \(1 + \sin^{2} x\) accordingly.
Simplify the integral to a rational function in \(u\) and then integrate with respect to \(u\). After integration, substitute back \(u = \tan x\) to express the answer in terms of \(x\).
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Trigonometric Substitution
Trigonometric substitution involves replacing a trigonometric function with a new variable to simplify an integral. In this problem, substituting u = tan x transforms the integral into a rational function of u, making it easier to integrate.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions
Pythagorean Identities
Pythagorean identities relate sine, cosine, and tangent functions, such as 1 + tan²x = sec²x. These identities are essential for rewriting expressions involving sin²x in terms of tan x and sec x, facilitating substitution and simplification.
Video consigliato:
7:17Verifying Trig Equations as Identities
Integration of Rational Functions
After substitution, the integral often becomes a rational function in terms of u. Understanding how to integrate rational functions, including techniques like partial fractions or recognizing standard forms, is crucial to solving the integral.
Video consigliato:
6:04Intro to Rational Functions
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