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Problem 82 Hard Difficulty

Show that if $ a > -1 $ and $ b > a + 1 $, then the following integral is convergent
$$ \int_0^\infty \frac{x^a}{1 + x^b}\ dx $$


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Related Topics

Integration Techniques

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01:53

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.

Video Thumbnail

27:53

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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Watch More Solved Questions in Chapter 7

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Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
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Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80
Problem 81
Problem 82

Video Transcript

The problem is show that if its greatest on ninety one and B is created on a past one, then the following into grow is converted. This improper integral we can, right? It's integral. Is he called to integral from zero to one ax to a over one plus X to be the eggs class into girl from one to the infinity of ax to a for one flat sachs to be power? Yes, on the first part is is a definite integral, so it will be a constant a number. So if the second heart is converted, then are integral from zero to infinity of dysfunction is commitment. Now look at this improper integral. We can write disfunction, eyes divided acts to its power for the numerator and the denominator. So this is one over one. Over. Ax to ace Power class Act two I mean minus a Yes. This function is always smaller, Czar, into your from one to infinity. One over x So e minus its power is it for this function? Denominator won over acts to ace power plus acts to humanise, eh is greeted on axe to be minus his power. These axes, critters and zero. Now look at this improper anti girl Face B is created on a past one. It's a B minus one. It's three to one. So this improper anti grow is commitment. And by the comparison, cereal thiss improper Anti Girl is also important is an integral from zero to infinity of the function axed weigh over one plus x two b's power is also converted.

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Related Topics

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Top Calculus 2 / BC Educators
Grace He

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Anna Marie Vagnozzi

Campbell University

Heather Zimmers

Oregon State University

Joseph Lentino

Boston College

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

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.

Video Thumbnail

27:53

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.

Join Course
Recommended Videos

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Consider the improper integral $$\int_{1}^{\infty} x^{-2 / 3} d x$$ a. Evaluate…

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