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

$ \displaystyle \int_{-\infty}^0 \frac{1}{3 - 4x}\ dx $

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$\infty,$ divergent

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Campbell University

Harvey Mudd College

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.

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Determine whether each int…

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Determine whether each imp…

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First, we can use substitution to convert this function to a function that is easier to integrate, that, u is equal to 3 minus 4 x, then d? U is equal to negative 4 times dx, then integral of this function is equal to 1. Over, u times negative 1 force du integral from infinity to 3, a d? U is equal to negative 4 times d x d x is equal to negative 14 d and 1 x goes to negative infinity. U goes to infinity and while x goes to 0. U goes to 3 point and then we can rewrite this integral to 14 times integral of 1 over. U, from 3 to infinity and negative integral from infinity to 3, is equal to integral from 3 to infinity point, then, by definition, this is equal to the limit. T goes to infinity of 1, he 4 times integral 1 over. U from 3 to t o. This is equal to the limit of t goes to infinity 14 times. L n t minus 3 point in 1 goes to infinity. N t goes to infinity, so the result is infinity, so this integral is deaden.

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