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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 8.7.24
Chapter 8, Problem 8.7.24

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
24. ∫ dt / √(1 + 4eᵗ)

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1
Step 1: Recognize that the integral ∫ dt / √(1 + 4eᵗ) does not directly match a standard form in the table of integrals. Begin by identifying a substitution that simplifies the expression. Let u = 4eᵗ, which implies du = 4eᵗ dt or dt = du / (4eᵗ).
Step 2: Substitute u = 4eᵗ into the integral. The integral becomes ∫ (1 / √(1 + u)) * (du / 4eᵗ). Since eᵗ = u / 4, replace eᵗ with u / 4 in the denominator, simplifying the integral to ∫ (1 / √(1 + u)) * (du / u).
Step 3: Simplify the integral further. The expression now becomes (1/4) ∫ (1 / (u√(1 + u))) du. This matches a standard form in the table of integrals, which involves the substitution u = x² or another transformation to simplify the square root.
Step 4: Refer to the table of integrals for the standard form that matches ∫ (1 / (u√(1 + u))) du. Use the appropriate formula from the table to rewrite the integral in terms of u.
Step 5: After applying the formula from the table, back-substitute u = 4eᵗ to express the result in terms of t. Ensure the final expression is simplified and represents the indefinite integral.

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Key Concepts

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they provide the antiderivative of a function.
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Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method is often necessary for simplifying integrals involving quadratic terms, making them easier to integrate. It allows for the identification of standard integral forms in tables.
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Completing the Square to Rewrite the Integrand

Integral Tables

Integral tables are collections of standard integrals that provide quick references for evaluating common integrals. They simplify the integration process by allowing students to look up known forms rather than calculating integrals from scratch. Familiarity with these tables is essential for efficiently solving complex integrals.
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Related Practice
Textbook Question

7–64. Integration review Evaluate the following integrals.

20. ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) dx

Textbook Question

5. What is a reduction formula?

Textbook Question

50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:

53. ∫ lnⁿ(x) dx = x lnⁿ(x) - n ∫ lnⁿ⁻¹(x) dx

Textbook Question

7–84. Evaluate the following integrals.

22. ∫ [1 / ((x - a)(x - b))] dx, where a ≠ b

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Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

54. ∫ y⁴/(1 + y²) dy

Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt