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Suppose $ f $ is continuous on $ [0, \infty) $ and $ \lim_{x\to\infty} f(x) = 1 $. Is it possible that $ \displaystyle \int_0^\infty f(x)\ dx $ is convergent?

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

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.

00:59

Suppose $f$ is continuous …

03:42

the problem is, suppose is continuous on interval from territory Infinity on the limit. My fax my ex goes to infinity is they want It seemed impossible that any girl from zero to infinity about axe is converted. The other is no. The reason is that's the definition of continuous function on bits of probably that affect the limits of our eyes. On Mexico's to invent is one. We have the following results. There must be a good list. Some number, eh? On some number I'm It's through this war on their own. Small one one x is greatest on, eh? The value of a fax is greatest on. Um no, this's a definition of continuous function on DH. It's the definition of limited backs. Goes to independently of our fax you one Now, Auntie girl from zero to infinity of fax The axe s s legal. Choose an integral from zero to a max. The axe us et tu impunity of fax the ex No. And this card it's a fax is continuous. This is a definite integral. There will be some constant number. It is a constant number on DH here, but this integral from eighty Infinity base one axe is correct and the creators are a fax is criticism. Um, So this part is readers up into girl from a to infinity, some on the X and this and stammered and yeah, so our function a fax from zero to infinity is also that gradient.

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