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

$ \displaystyle \int_2^\infty ye^{-3y}\ dy $

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convergent to $\frac{7}{9 e^{-6}}$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

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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.

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Determine whether each int…

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The problem is determine whether each integral is converted or divergent and evaluated those that are converted for this improper, integral a definition equal to the limit. A goes to infinity of integral 2 to a y times e to negative 3 y d y. And then we computed the definite integral first, this definite integral. We is the method of the integration by parts. This is equal to integral from 2 to a y d, 1 or negative. 1 third have e to the negative 3 y. So this is equal to y times negative 1 third in to negative 3 y from 2 to a minus integral of 2 to a negative 1 third to negative 3 y dy. This is equal to y times negative 1 third e to negative 3 y from 2 to a and the integration of the function negative 1 third times e to negative 3 y is equal to 1 over 9 into negative 3 y. This is from 2 to a andalugian 2. These 2 functions the distance equal to a times negative 1 over 3 e to negative 3, a minus 2 times negative 1 over 3 e to negative 6, and the sin is minus 1, over 9 e to negative 3. A minus e to negative 6, so this i go to a sum. 8 goes to infinity. This part goes to 0 and this part also goes to 0 point. The answer should be equal to 2 third plus 1 over 9 times e to negative 6, or this is equal to 7 over 9 times e to negative 6 point. So this improper integral is converted and the value is 7 or 9 ti negative 6.

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