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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 8.7.15
Chapter 8, Problem 8.7.15

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.
15. ∫ x / √(4x + 1) dx

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Rewrite the integral in a form that is easier to match with a table of integrals. The given integral is ∫(x / √(4x + 1)) dx. Notice that the denominator involves a square root, so we will aim to simplify the expression.
Perform a substitution to simplify the square root. Let u = 4x + 1, which implies that du/dx = 4 or dx = du/4. Also, note that x = (u - 1)/4 from the substitution.
Substitute x and dx into the integral. The integral becomes ∫[((u - 1)/4) / √u] * (1/4) du. Simplify this expression to ∫[(u - 1) / (16√u)] du.
Split the integral into two simpler terms: ∫[(u / (16√u))] du - ∫[(1 / (16√u))] du. Simplify each term. The first term becomes (1/16)∫√u du, and the second term becomes (1/16)∫u^(-1/2) du.
Use the power rule for integration to evaluate each term. For the first term, ∫√u du = ∫u^(1/2) du = (2/3)u^(3/2). For the second term, ∫u^(-1/2) du = 2u^(1/2). Combine the results and substitute back u = 4x + 1 to express the solution in terms of x.

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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, typically denoted as 'C'. Understanding how to compute indefinite integrals is fundamental in calculus, as it allows for the determination of antiderivatives, which are essential in solving differential equations and analyzing functions.
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Introduction to Indefinite Integrals

Table of Integrals

A table of integrals is a reference tool that lists common integrals and their corresponding antiderivatives. It simplifies the process of finding integrals by providing ready-made solutions for frequently encountered functions. Familiarity with this table can significantly expedite the evaluation of integrals, especially when combined with techniques like substitution or completing the square.
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Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is particularly useful in integration, as it can simplify the integrand, making it easier to apply integration techniques or look up in a table. For example, rewriting expressions in the form of (x + a)² helps in recognizing standard integral forms.
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Related Practice
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

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

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78. ∫(from 0 to ∞) dx / (eˣ + x + 1)

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Textbook Question

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

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