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Evaluate the integral.
$ \displaystyle \int \frac{x^2}{x^6 + 3x^3 + 2}\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 5
Strategy for Integration
Integration Techniques
University of Michigan - Ann Arbor
Idaho State University
Boston College
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 given integra…
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Evaluate the integrals.
Let's start off here with the use of Let's take you to be x cubed, then do you? Over three is X square D X and that's exactly what we see in the numerator over here. X squared dx. So also, we have to rewrite this exit six is X cubed square so that you square so on the bottom. Let's do the denominator first. Excellent. Six three x cubed plus two in on the top where he saw that this is just do you and then we have to divide by the three Let's pull off the one third. So now this is a much better looking at a girl because weaken factor that denominator And then we could do partial fraction to composition here. So in this case, you want to find numbers and be such that we can rewrite the immigrant to be this, and it turns out that we have b who's won. So we have won over you, plus one and then we have a is minus one. All right, so use partial fraction the composition match up the coefficients to find your A and B as we did in the previous section. God and evaluate thes. That's a three out there, plus one minus. It's one thing I heard and then plus two. So here I just distributed the one third and generated both of these. I added that constancy of integration. So here we could do two steps. So pull out the winter and then I rewrite the longer of them as a fraction because its attraction and then I just replace you with execute. So on top of you, plus one so X cubed, plus one and then on the bottom X cubed, plus two. And then let's add in our constant of integration, see? And there's our final answer.
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