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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 8.5.79
Chapter 8, Problem 8.5.79

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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1
Start by examining the integral \( \int \frac{\sec t}{1 + \sin t} \, dt \). Notice that the denominator contains \(1 + \sin t\), which suggests a trigonometric identity or substitution might simplify the expression.
Recall the Pythagorean identity and the conjugate expression: multiply numerator and denominator by the conjugate of the denominator, which is \(1 - \sin t\), to simplify the denominator using the difference of squares formula.
Rewrite the integral as \( \int \frac{\sec t (1 - \sin t)}{(1 + \sin t)(1 - \sin t)} \, dt = \int \frac{\sec t (1 - \sin t)}{1 - \sin^2 t} \, dt \).
Use the Pythagorean identity \(1 - \sin^2 t = \cos^2 t\) to simplify the denominator, so the integral becomes \( \int \frac{\sec t (1 - \sin t)}{\cos^2 t} \, dt \).
Express \(\sec t = \frac{1}{\cos t}\) and rewrite the integral as \( \int \frac{1 - \sin t}{\cos^3 t} \, dt \). From here, consider a substitution such as \(u = \sin t\) to proceed with integration.

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They help simplify expressions and integrals by rewriting functions in more convenient forms, such as converting secant and sine into cosine and sine or using Pythagorean identities.
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Verifying Trig Equations as Identities

Substitution Method (Change of Variables)

The substitution method involves changing the variable of integration to simplify the integral. By choosing an appropriate substitution, the integral can be transformed into a more manageable form, often turning complicated expressions into rational functions suitable for further techniques like partial fractions.
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Substitution With an Extra Variable

Partial Fraction Decomposition

Partial fraction decomposition breaks down a complex rational function into simpler fractions that are easier to integrate. This method is especially useful after substitution when the integral becomes a rational function, allowing straightforward integration of each simpler term.
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Partial Fraction Decomposition: Distinct Linear Factors
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

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Textbook Question

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.

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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.

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Textbook Question

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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Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

25. ∫ sin²x cos⁴x dx