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Evaluate the integral. $\int \frac{x-1}{x^{2}+2 x} d x$

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02:50

Wen Zheng

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

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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. $\i…

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Evaluate the integral.…

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So here we are given examples of certain integral and this is in the form of partial fraction decomposition. So you have to do that process. Yeah. Which is equivalent to a over X plus B over X plus two or A and D are constants. Mhm Okay, so let's multiply both sides by X and x plus two. So this would be x minus one equals A times X plus two plus B times x. So let's group terms together. So A plus B, X plus two, A plus B. So we can see by just coefficients, A plus B is equivalent to one to A plus B equals negative one. So negative A minus B equals negative one. So A is equivalent to negative choose. So it is equivalent to negative to be would be equivalent to three in this case. So now we can apply our integral directly. So a was negative negative two. So negative two over X. Which is just a constant plus three over X plus two T X. So this is equivalent to negative two natural log of X Plus three. Natural log of a. Yeah exposed to. So this would be the same thing as the natural log of X to the -2. Which would be X In this form plus three natural log of X plus two. Or we can write as the que perform plus our integration constant. And this gives our final answer

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