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(a) If $ g(x) = \frac{1}{(\sqrt{x} - 1)} $, use your calculator or computer to make a table of approximate values of $ \displaystyle \int_2^t g(x)\ dx $ for $ t = 5, 10, 100, 1000 $, and $ 10,000 $. Does it appear that $ \displaystyle \int_2^\infty g(x)\ dx $ is convergent or divergent?(b) Use the Comparison Theorem with $ f(x) = \frac{1}{\sqrt{x}} $ to show that $ \displaystyle \int_2^\infty g(x)\ dx $ is divergent.(c) Illustrate part (b) by graphing $ f $ and $ g $ on the same screen for $ 2 \le x \le 20 $. Use your graph to explain intuitively why $ \displaystyle \int_2^\infty g(x)\ dx $ is divergent.

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

04:59

(a) If $g(x)=1 /(\sqrt{x}-…

02:22

04:30

04:26

(a) If $ g(x) = \frac{(\si…

06:08

(a) If $g(x)=\left(\sin ^{…

0:00

03:43

02:54

Consider the functions $f(…

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