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Evaluate the integral by completing the square and using Formula 6.

$ \displaystyle \int \frac{dx}{x^2 - 2x} $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Harvey Mudd College

University of Nottingham

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.

03:33

Evaluate the integral by c…

06:07

06:29

01:40

Evaluate the integrals.

04:19

Use the technique of compl…

04:03

Evaluate the integrals by …

01:06

Let's evaluate the integral by completing the square and using Formula Six. So I've written Formula Six here on the right from yourself. Book. Now it's completely square civil. Add one and then subtract one. We have X minus one Swear minus one. If we want with. Let's also right. This's once were due to the fact that we have a sweat over here on the right. So now we could rewrite this in a girl DX X minus, one squared, minus one square. And if that minus one is throwing you off because it's not showing up over here and it's up right, I could go ahead and do a use up here. U equals X minus one. Do it was the ex. Then this in rule, becomes do you use square minus one's where and now we use the formula here a equals one. So then we have one over two times one and then there We should not have an interval any more. On the right hand side, we're using Formula Six over here, So one over two times one natural log and then we have X minus a and then X plus a. So here we have you minus four. You plus one. And then we add Si. And then now we just go ahead, replace you with excusing this. So we have one half, and then we have X minus two over X plus our constancy, and that will be our final answer.

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