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Find the value of the constant $ C $ for which the integral$$ \displaystyle \int_0^\infty \left (\frac{x}{x^2 + 1} - \frac{C}{3x + 1} \right)\ dx $$converges. Evaluate the integral for this value of $ C $.

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Oregon 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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Finding a Value For what v…

the problem is finding one year with the constant number See awaits the integral interpret from zero to infinity of this function murders The water is in the girl for this model? No. For this improper, integral attacked nation they safety conscious limit A goes to infinity of integral from zero to a act's over square plus one Yes, minus the girl from zero to a c o r. Reax Ross Long Yes. Now first one to go from zero to a ax over square us. Lan Yes. This is a two here. We use use substitution, not you is equal to x square. The U. S is going to two eggs? Yes. Now it's integral is equal to I'm a hero to a square. One half you over you plus one. This's the go to l A and you plus one from zero to a square. Yeah, One half. I'm planning a square and zero dysfunctions, which this is, you know, and a square off one on the weekend, Right? One half here is my house. Oh, on the second heart into girl from zero to a see over three X plus one. Yes, this is the contusions they are three. Oh, and reax swan from zero to A This is a code to, um three A plus one two. They are three spow er now the into girl from zero to infinity X over X square. That's Juan minus. They are reacts a swan. Jax. They got to the limit. A ghost now and a square US one one over. Why us three? A us one? Yeah, they over three. Power. This is the culture Lim a goes to infinity. Ellen a square us one one. Uh oh. Three a. Teo Theo are Reese power. Look at the degree off a photo. The denominator on numerator But the numerator degree of is going to want. So if the limit exist, degree of the number off the denominator must be one. So see over rain should be equal to what? And we have. Is he going to three? So you're saying they got three limit? You consist on DH. This's e kowtow our if c is equal to three. This is three a plus. Wantto wants power. It is going to one over. Ray, this's a while off this anti girl

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