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Evaluate the integral.

$ \displaystyle \int x^3 (x - 1)^{-4}\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Campbell University

Harvey Mudd College

Baylor University

University of Nottingham

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.

03:46

Evaluate the integral.…

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02:04

Evaluate the definite inte…

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03:48

Let's use a U substitution. Here, let's take you. It's just B X minus one motivated by this term here with the negative exponents. Then do you equals the X? So here we can come and rewrite. This of the first term is execute. So first off this rex, so x cubed becomes you plus one cube and then we have you to the minus four so we can rewrite. This is you plus one cute. You know the four and then the next step here we can just go ahead and expand that numerator. So go ahead and just cube this you plus one. And then the next step here would be to just split this up into foreigner girls. So for fractions here for the first fraction, we just have one over you Next. Next Fraction three, let's write. This is Yoo to the minus two. And then for the next one three year, the minus three. And for the last one, you to the minus four. So here we just use the power rule four times. First in the rule that's natural log absolute value, and then next will have negative one over negative three over you Excuse me, then minus three over to you squared. And then he's here for the last one. You'd have negative one over three you killed. So, sir, Constants of integration. See, the last step here is just to replace all of the EU's with X minus one. So we have will do. This is our last step. So that's the use Where? Right there and in minus one over three. And then you cube here, plus our constancy, and that's your final answer.

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