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

$ \displaystyle \int_0^\infty \frac{1}{\sqrt[4]{1 + x}}\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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 this integral is converted or their warden. This is an improper integral by definition, this is equal to the limit, goes to infinity and integral of 1 over 1 plus x, 21 fourth power from 0 to t dexo. We can rewrite this function as limited goes to infinity integral from 0 to t this is 1 plus x to negative 1 force power tex. Then we need to find anti derivative of this function. This is equal to it. It goes to infinity integral anti derivative. With this function is 1 over minus 1 over 4 plus 11 plus x 2 negative 1 force has long from 0 to t, then plotting key and there here this is equal to the limit. He goes to infinity. This is 1 over 3 over 4 point. So this is, this is 4 over 3 and this is y plus t 3 over 4 minus 43 times 1 face to the power of 1 plus t is positive, 1 t goes to infinity 1 plus t 23 or foure power goes to infinity. So the answer is infinity. This integral is dearden.

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