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Problem

Use the Comparison Theorem to determine whether t…

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Problem 48 Hard Difficulty

(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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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

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Improper Integrals

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Integration Techniques - Intro

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.

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Basic Techniques

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

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Video Transcript

for a If Jax is you, Goto won over root of ax minus one. Use your calculator. Our computer to make a table of approximate value of the integral of Jack's from two to tea or tears, they goto five ten hundred thousand. Ten thousand. That's it appears that improper, integral the jukebox is combatant or de evident. Monty is equal to five turn one hundred, one thousand and ten thousand a computation. Antero is equal to three point eight three or three. Six point eight o one two. Twenty three point three two eight eight. I'm sixteen. Sixty nine on there are two, three, four and two hundred and eight point one two for a fix. It seems that in improper, integral as that wouldn't since twenty is got at bigger and bigger into girl. Coming very large will be We know what Jill Bugs is Creatures on one over your axe things the denominator Rooter vax minus one is smaller. That toe axe. So, Auntie Girl off the box as narrator inthe Ural of Ethel box over Conrad that one over Roots of X, Jax. But we know this integral establishment things. The power of axe is about half power of axe is my half. It's more a czar one, So this integral is evident. That comparison Siri. Um so the improper integral of jewel box as also verdant. Oh, Patsy, look at the graph here. The blue on is Jill Box and is a writer one. It's half a box we can't see area on the it's to curves. So under the blue one is you guys are every year and there's a right away says into girl off jukebox as greed has on into girl of half a box and use your graft explain intuitively why integral of Jukebox s tavern And so we can see area and there's a blue one One key got a lot and larger Just relax. It's not worry us more so we can imagine area Aunt Bea by re large when he goes to infinity

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Video Thumbnail

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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.

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