Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.1.7

Chapter 8, Problem 8.1.7

7–64. Integration review Evaluate the following integrals.
7. ∫ dx / (3 - 5x)^4

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Recognize that the integral ∫ dx / (3 - 5x)^4 involves a rational function with a power of a linear term in the denominator. This suggests using a substitution to simplify the integral.

Let u = 3 - 5x. Then, compute the derivative of u with respect to x: du/dx = -5, or equivalently, dx = -du/5.

Substitute u and dx into the integral. The integral becomes ∫ (-1/5) du / u^4, where the factor -1/5 comes from the substitution for dx.

Simplify the integral to ∫ -1/5 * u^(-4) du. Rewrite the integrand as -1/5 * u^(-4) to prepare for applying the power rule of integration.

Apply the power rule for integration: ∫ u^n du = u^(n+1) / (n+1), for n ≠ -1. Here, n = -4, so the integral becomes (-1/5) * (u^(-4+1) / (-4+1)). Simplify the expression and substitute back u = 3 - 5x to return to the original variable.

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Key Concepts

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

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fractions. In this case, substitution is particularly useful for simplifying the integral of a rational function, allowing for easier evaluation.

Substitution Method

The substitution method involves replacing a variable in the integral with another variable to simplify the expression. For the integral ∫ dx / (3 - 5x)^4, we can let u = 3 - 5x, which transforms the integral into a more manageable form. This method is essential for integrals involving composite functions.

Power Rule for Integration

The power rule for integration states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1. This rule is applicable when integrating functions of the form u^n, where u is a function of x. In the given integral, after substitution, applying the power rule will help in finding the antiderivative of the transformed function.

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Related Practice

Textbook Question

59. Area of a segment of a circle

Use two approaches to show that the area of a cap (or segment) of a circle of radius r subtended by an angle θ (see figure) is given by:

A_seg = (1/2) r² (θ - sin θ)

b. Find the area using calculus.

Textbook Question

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

15. x / ((x⁴ - 16)²)

Textbook Question

23-64. Integration Evaluate the following integrals.

35. ∫ (x² + 12x - 4)/(x³ - 4x) dx

Textbook Question

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

30. ∫ x³√(1 - x²) dx

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

13. ∫ sin⁵x dx

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.