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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.1.38
Capitolo 8, Problema 8.1.38

7–64. Integration review Evaluate the following integrals.
38. ∫ x / (x⁴ + 2x² + 1) dx

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Step 1: Observe the integrand ∫ x / (x⁴ + 2x² + 1) dx. Notice that the denominator can be factored. Rewrite the denominator as (x² + 1)², since x⁴ + 2x² + 1 is a perfect square trinomial.
Step 2: Substitute u = x² + 1 to simplify the integral. Compute the derivative of u with respect to x: du/dx = 2x, which implies du = 2x dx.
Step 3: Rewrite the integral in terms of u. Substitute x dx with (1/2) du, and the denominator becomes u². The integral now becomes (1/2) ∫ 1/u² du.
Step 4: Apply the power rule for integration to ∫ 1/u² du. Recall that ∫ u⁻² du = -u⁻¹ + C, where C is the constant of integration.
Step 5: Substitute back u = x² + 1 into the result to express the solution in terms of x. The final answer will be in the form of -1/(x² + 1) + C.

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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 more complex integrals, such as the one presented in the question.
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Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers. In the integral ∫ x / (x⁴ + 2x² + 1) dx, the denominator is a polynomial of degree four. Recognizing the structure of polynomial functions helps in simplifying the integral and determining appropriate integration techniques.
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Rational Functions

Rational functions are ratios of two polynomial functions. The integral in the question involves a rational function, which can often be simplified or decomposed for easier integration. Understanding how to manipulate rational functions is essential for effectively evaluating integrals like the one given.
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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.

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23-64. Integration Evaluate the following integrals.

35. ∫ (x² + 12x - 4)/(x³ - 4x) dx

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Evaluate the following integrals.

∫ x/(x² + 6x + 18) dx

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65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

75. The region bounded by f(x) = (4 - x)^(-1/3) and the x-axis on the interval [0, 4) is revolved about the y-axis.

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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.

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7–84. Evaluate the following integrals.

41. ∫ cot^(3/2)x · csc⁴x dx

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