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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.5.96
Capitolo 8, Problema 8.5.96

96. Challenge
Show that with the change of variables u = √tan x, the integral
∫ √tan x dx
can be converted to an integral amenable to partial fractions. Evaluate
∫[0 to π/4] √tan x dx.

Guida verificata passo dopo passo
1
Start with the substitution given: let \( u = \sqrt{\tan x} \). This means \( u^2 = \tan x \).
Differentiate both sides with respect to \( x \) to find \( du \) in terms of \( dx \): \( \frac{d}{dx}(u^2) = \frac{d}{dx}(\tan x) \) which gives \( 2u \frac{du}{dx} = \sec^2 x \). Solve for \( dx \) in terms of \( du \) and \( u \).
Express \( \sec^2 x \) in terms of \( u \) using the identity \( \tan^2 x + 1 = \sec^2 x \). Since \( \tan x = u^2 \), then \( \sec^2 x = 1 + u^4 \).
Rewrite the integral \( \int \sqrt{\tan x} \, dx = \int u \, dx \) by substituting \( dx \) from step 2 and \( \sec^2 x \) from step 3, to get the integral entirely in terms of \( u \) and \( du \).
Simplify the resulting integral to a rational function in \( u \) that can be decomposed into partial fractions. Then set up the partial fraction decomposition to prepare for integration.

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Concetti chiave

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Change of Variables (Substitution)

This technique involves replacing the original variable with a new one to simplify the integral. By expressing the integral in terms of a new variable, complex expressions can become easier to handle, often transforming the integral into a more familiar or solvable form.
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04:27
Substitution With an Extra Variable

Partial Fraction Decomposition

Partial fractions break down a complex rational function into simpler fractions that are easier to integrate. This method is especially useful when the integrand is a rational expression, allowing the integral to be expressed as a sum of simpler terms.
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Percorso guidato
10:07
Partial Fraction Decomposition: Distinct Linear Factors

Definite Integration with Trigonometric Limits

Evaluating definite integrals involving trigonometric functions requires careful substitution and adjustment of limits. When changing variables, the limits must be transformed accordingly to maintain the integral's value over the specified interval.
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Percorso guidato
05:43
Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

79. ∫ [sec t / (1 + sin t)] dt

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6. Using the trigonometric substitution x = 8 sec θ, where x ≥ 8 and 0 < θ ≤ π/2, express tan θ in terms of x.

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27. {Use of Tech} Midpoint Rule, Trapezoid Rule, and relative error

Find the Midpoint and Trapezoid Rule approximations to ∫(0 to 1) sin(πx) dx using n = 25 subintervals. Compute the relative error of each approximation.

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3. Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.

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60–69. Completing the square Evaluate the following integrals.

68. ∫ dx / sqrt((x - 1)(3 - x))

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87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

91. Evaluate ∫[0 to π/2] dθ/(cos θ + sin θ).

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