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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.5.96
Chapter 8, Problem 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.

Verified step by step guidance
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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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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Definition of the Definite Integral
Related Practice
Textbook Question

"Electric field due to a line of charge A total charge of Q is distributed uniformly on a line segment of length 2L along the y-axis (see figure). The x-component of the electric field at a point (a, 0) is given by

Eₓ(a) = (kQa/2L) ∫-L L dy/(a² + y²)^(3/2),

where k is a physical constant and a > 0.

a. Confirm that Eₓ(a)=kQ / a √(a²+L²)

b. Letting ρ=Q / 2 L be the charge density on the line segment, show that if L → ∞, then Eₓ(a) = 2kρ / a.

Textbook Question

29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

30. ∫(0 to 6) (x³/16 - x) dx = 4

1
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Textbook Question

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

Textbook Question

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

1
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Textbook Question

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.

82. ∫ [dx / (x√(1 + 2x))]

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

25. ∫ sin²x cos⁴x dx