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Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^\infty e^{-\sqrt{y}}\ dy $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Campbell University

University of Nottingham

Idaho State University

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.

04:44

Determine whether each int…

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Determine whether the inte…

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05:13

01:05

01:37

01:48

04:47

03:38

02:17

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05:01

03:44

03:29

The problem is determine why the integral is converted or abordent evaluated source that converted this improper integral a definition. This is equal to the limit goes to infinity and the integral is from 0 to t e to negative root of a we compute, the definitely integral. First. For this definite integral, we can use u substitution first, that? U is equal to negative root of a? U? U square is equal to y e. 2? U! D! U is equal to d y now the stephe integral is equal to 0 to negative root of t, and this is e to? U times? 2? U d! U here we use the method of the integration by postis is equal to integral from theory to negative root of, and this is 2? U d? U t! This is equal to 2! U e t u, from 0 to negative root of minus integral from 0 to negative root of t of the function 2 times e to u du! This is equal to negative root of t times 2 negative 2 times the root of t e to negative root of t minus 0. Minus and anti derivative of this function is 2 times a to u, from 0, to negative root of this equal to negative 2 times root of t e to negative root of t minus 2 times e to negative root of minus 2. When t goes to infinity, this part goes to 0 and this part also goes to 0. So the answer is equal to 2 point. So this improper, integral is converteth. Value is 2.

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