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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 8.9.98
Chapter 8, Problem 8.9.98

95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.
98. ∫(from 0 to 1) (ln x)/(1+x) dx = -π²/12

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1
Recognize that the integral to approximate is \(\int_0^1 \frac{\ln x}{1+x} \, dx\), which is improper at the lower limit since \(\ln x\) tends to \(-\infty\) as \(x \to 0^+\).
Choose a suitable numerical integration method, such as Simpson's Rule, Trapezoidal Rule, or Gaussian Quadrature, to approximate the integral. Because of the singularity at 0, consider using a method that handles improper integrals or apply a substitution to manage the singularity.
If using a substitution, for example, set \(x = t^2\) or another transformation to smooth the integrand near 0, then rewrite the integral in terms of the new variable and adjust the limits accordingly.
Divide the interval \([0,1]\) into a number of subintervals (the more subintervals, the better the approximation) and calculate the function values at the chosen points according to the numerical method selected.
Sum the weighted function values according to the numerical method's formula to approximate the integral, and compare the result to the exact value \(-\frac{\pi^2}{12}\) to check the accuracy.

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Key Concepts

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

Numerical Integration

Numerical integration involves approximating the value of definite integrals when an exact analytical solution is difficult or impossible. Common methods include the Trapezoidal Rule, Simpson's Rule, and Gaussian Quadrature, which estimate the area under a curve using discrete sample points.
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Handling Improper Integrals

Improper integrals occur when the integrand is undefined or unbounded at some points in the interval, such as ln(x) at x=0. Special techniques, like limit processes or variable substitution, are used to evaluate or approximate these integrals accurately.
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Use of Technology in Calculus

Calculators and software tools can perform numerical integration efficiently, especially for complex or improper integrals. Understanding how to input functions and interpret outputs from these tools is essential for verifying results and handling integrals that are difficult to solve analytically.
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Related Practice
Textbook Question

17-22. Give the partial fraction decomposition for the following expressions.

17. (5x - 7) / (x² - 3x + 2)

Textbook Question

7. How would you evaluate ∫ tan¹⁰x sec²x dx?

Textbook Question

7–84. Evaluate the following integrals.

16. ∫ [1 / (x⁴ – 1)] dx

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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.

31. ∫ √(x² - 8x) dx, x > 8

Textbook Question

63. (Use of Tech) Normal distribution of heights

The heights of U.S. men are normally distributed with a mean of 69 in and a standard deviation of 3 in. This means that the fraction of men with a height between a and b (with a < b) inches is given by the integral

(1/(3√(2π))) ∫ₐᵇ e^(-((x-69)/3)²/2) dx.

What percentage of American men are between 66 and 72 inches tall? Use the method of your choice, and experiment with the number of subintervals until you obtain successive approximations that differ by less than 10⁻³.

Textbook Question

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

91. Evaluate ∫[0 to π/2] dθ/(cos θ + sin θ).