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Evaluate the integral.

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

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Campbell University

Harvey Mudd College

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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Evaluate the integral.…

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Evaluate the integral. $\i…

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02:37

let's evaluate the integral of X divided by X squared minus B X. We see that there's a quadratic in the bottom in that denominator, so always look to see if that could be factored here. Fortunately, we could take another X and then in this case, there's no need for the partial fraction to composition because we could just go ahead and cancel the exes. And we just have a over X minus B. So if you want, this is R and read, this is our partial fraction to composition. We just have a constant over a linear term. Explain his B So always factor when you can doesn't happen all the time, but it could make the problem easier. Now, at this point, we have an integral that you've seen before. If this X minus B is was throwing you off in the denominator, they'd just go ahead and take a use up. So let's go ahead and do this Use up and that are in rule becomes a over you and then D s. It's just do you and we know this to be eight times natural log absolute value of you. Don't forget the constancy of integration and then, at this point, just back substance. Who you for X minus B a natural log X minus B plus he and there's a final answer. Go.

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