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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.R.18
Chapter 8, Problem 8.R.18

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
18. ∫ (from 0 to √2) (x + 1)/(3x² + 6) dx

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Step 1: Simplify the integrand. Start by factoring the denominator: \(3x^2 + 6 = 3(x^2 + 2)\). Rewrite the integrand as \(\frac{x + 1}{3(x^2 + 2)}\).
Step 2: Split the integrand into two separate terms: \(\frac{x}{3(x^2 + 2)} + \frac{1}{3(x^2 + 2)}\). This allows us to handle each term individually.
Step 3: For the first term \(\frac{x}{3(x^2 + 2)}\), notice that the numerator is the derivative of the denominator \(x^2 + 2\). Use the substitution method: let \(u = x^2 + 2\), then \(du = 2x dx\). Adjust for the constant and rewrite the integral.
Step 4: For the second term \(\frac{1}{3(x^2 + 2)}\), recognize it as a standard integral form. Use the formula \(\int \frac{1}{a^2 + x^2} dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right)\), where \(a = \sqrt{2}\). Apply this formula to evaluate the integral.
Step 5: Combine the results of both integrals and evaluate the definite integral by substituting the limits \(x = 0\) and \(x = \sqrt{2}\). Simplify the expression to obtain the final result.

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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 fraction decomposition. Understanding these methods is crucial for evaluating complex integrals, as they allow for simplification and easier computation.
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Definite Integrals

A definite integral calculates the area under a curve between two specified limits. It is represented as ∫ from a to b f(x) dx, where 'a' and 'b' are the bounds of integration. Evaluating definite integrals involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus.
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Definition of the Definite Integral

Rational Functions

Rational functions are ratios of polynomials, expressed as f(x) = P(x)/Q(x), where P and Q are polynomials. When integrating rational functions, techniques such as polynomial long division and partial fraction decomposition are often employed to simplify the integrand, making it easier to integrate.
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Related Practice
Textbook Question

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

b. To evaluate the integral ∫dx/√(x² − 100) analytically, it is best to use partial fractions.

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Textbook Question

114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).

a. Find the length of the curve from x = 1 to x = a and call it L(a).

(Hint: The change of variables u = sqrt(x^2 + 1) allows evaluation by partial fractions.)

Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

3. ∫ (3x)/√(x + 4) dx

Textbook Question

123. Region between curves Find the area of the region bounded by the graphs of y = tan(x) and y = sec(x) on the interval [0, π/4].

Textbook Question

76-81. Table of integrals Use a table of integrals to evaluate the following integrals.

76. ∫ x(2x + 3)⁵ dx

Textbook Question

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

d. ∫2 sin x cos x dx = −(1/2) cos 2x + C.

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