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

$ \displaystyle \int_{-\infty}^\infty xe^{-x^2}\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Missouri State University

Campbell University

Idaho State University

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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For this integral, we can write it as negative infinity to 0 as a function x, 2 times e to negative x, squared dx 0 to infinity of this function x times e to negative x, squared dx, we computed the first integral this is equal to this 1. This is equal to the limit as to negative infinity of the integral a 20 x times e to negative x, squared dx, and we would have computed the danite integral first and this part of this part. This is equal to here. We use substitution that, u equal to x square, then this is equal to e 2 negative. U times, 1 half u from a square to 0, and this is equal to 1 half times negative e to the negative? U from a square to? U point by computation! This is equal to negative 1 half times 1 minus e to negative a square, and this this function goes to negative 1 half when a goes to negative infinity, so the first part is equal to negative 1 half pint. Similarly, when an computed, the second part- and the second part is equal to 1 half, so the answer is equal to 0 point to this function. Integral this function is converted and the answer is 0.

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