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Evaluate the integral.

$ \displaystyle \int \frac{x^4 + 9x^2 + x + 2}{x^2 + 9}\ dx $

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$$\frac{1}{3} x^{3}+\frac{1}{2} \ln \left(x^{2}+9\right)+\frac{2}{3} \arctan (x / 3)+C$$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Missouri State University

Campbell University

Oregon State University

Harvey Mudd 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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Find the integral of x^3 /…

Let's evaluate to given integral. The first thing to observe here is that the numerator has larger degree than the denominator. So we should do polynomial division to simplify before we even consider a partial fractions. So let's do the division here exit the fourth divided by X squared. That's just ex clear. It's going well. Supply that out. Then we subtract and we're left over with X Plus two because we have cancellation here. So this tells us that we can rewrite the integral as that's a plus two. Excuse me, X Square. This is the question, and then we have our remainder X plus two over the original denominator X squared plus nine. So this is a simpler looking in a world in the original. There's really no need for partial for actions here because this term over here already is a partial fraction. We have a quadratic in the denominator that does not factor, and then you have your X plus B in the numerator. So really, we're ready to integrate. Let's break this into three of the rules, and then we could split up this fraction. This is just X over X squared plus sign, plus two over X Square plus nine. So it's got an integrated each of those. And then for the last one, it's good. And pull out that too. And now we have three in a girls. So the first one just use the power rule. We know that's just X cubed over three. Don't worry about the sea for right now. Will add that in later. Now, for the second integral, we should do it yourself. Here, let's take you to be X squared plus sign then do you over too equals X t X. So this is just one half in a girl, one over you deal, which is one half natural log absolute value. You equals one half natural log and then replace you with X square plus nine. And you could drop the absolute value if you want, because X square plus nine is positive. So that's it to up there in the exponents Me rewrite this and then we have one more inside girls ago. We have this integral and then times two for this integral Here we should go ahead and do it shrinks up. You could take X to be to be tan data. So that DX is three seek and square data debater And then this integral just becomes too. Then on top we have three c can't square on the denominator will have nine can square data plus nine We're going in fact, throughout that nine and then we use the fact here that's hand squared plus one. It's just he can't swear so we could cancel the sea Can't square terms Then we'LL have to over three in a girl the tangents cancel. So are the sea cans. Cancel. We just have the data that's just too over three theater and then data can be solved by the original streets of by solving for data. Divide both sides by three they didn't take the ark cannot decide And then that will cancel the change it on the left And then on the right hand side you have tan in verse X over three So we've evaluated all three in a girls. Now we just add our answers together. So we'LL have in several equals here. This was the first term this was the second term and here was the third term to over three and then we have ten inverse x over three Do not forget the constant CIA veneration. Let's go to the next page and write this answer first in a girl we had executed over three. Then for the next one. One has natural log, X squared plus sign. And then for the last one, the trick substitution. We have to over three thinner and that became ten inverse of X over three. And then finally, we can add a constant of immigration and that'LL be our answer.

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