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Explain why each of the following integrals is improper.
(a) $ \displaystyle \int_1^2 \frac{x}{x - 1}\ dx $(b) $ \displaystyle \int_0^\infty \frac{1}{1 + x^3}\ dx $(c) $ \displaystyle \int_{-\infty}^\infty x^2 e^{-x^2}\ dx $(d) $ \displaystyle \int_0^{\frac{\pi}{4}} \cot x\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 8
Improper Integrals
Integration Techniques
Missouri State University
Oregon State University
University of Michigan - Ann Arbor
University of Nottingham
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.
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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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Explain why each of the following integrals is improper. So before we do this problem, let's record the definition of improper integrals for function, f, x, integral of f x, a to b is improper. If interval a to b is infinite or f x has an infinite discontinuity from a to b. Now, let's do the problem for b and c the interval is infinite, so they are improper integrals for a while x goes to 1. The function x over x, minus 1 goes to infinite infinity because the denominator goes to 0, so this function has an infinite. This continuity of 1 to 2 point, so it is improper integral and for the 1 x, goes to 0 for 10 x goes to infinity, so the function of co tangent x has an infinite discontinuity among 02 pi over 4. So it is improper, integral.
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