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Write out the form of the partial fraction decomp…

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Problem 5 Medium Difficulty

Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients.

(a) $ \dfrac{x^6}{x^2 - 4} $ (b) $ \dfrac{x^4}{(x^2 - x + 1)(x^2 + 2)^2} $


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

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Integration Techniques

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01:53

Integration Techniques - Intro

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.

Video Thumbnail

27:53

Basic Techniques

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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Watch More Solved Questions in Chapter 7

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75

Video Transcript

Let's go ahead and find the parcel. Fraction the composition for these fractions given in party and be so let's start with party except the six Santa X squared minus four on bottom. The first thing that always check for this type of problem which by this problem, I mean polynomial divided by polynomial. We always want to know if the numerator has degree greater than or equal to the denominator. If it does, we should use long division here. Numerator has degree six Denominator has degree too. So we go for the long division. So it's going to the right and do this. We have exit the fourth over here and then I'LL give us extra sixth minus for us to the fourth power Go ahead and subtract And then up here we get a four x square minus sixteen x where no, subtract that we get sixteen x square And then that'LL give us a plus sixteen up here So that will give us sixteen X square minus sixty four so suppressed that ever left over with sixty four. So that's our manor suffer party. We can rewrite this long division give us quotient of X to the fourth for X Square plus sixteen. Then we have our remainder. And then we have the original denominator explored minus four. And now we just go ahead and use partial fractions on this on ly remaining fraction over here. And now we know no more long division will be needed because the nominator has been a degree. So we should take this denominator and go ahead and write that as X minus two always want to check if we can factor these and then we have two distinct linear factors. So this will be what the book caused. Case one. So we'LL use the formula for case one to rewrite the fraction and this will become. We have a over X minus two and then another constant be over X plus two and that will be an answer for party. Let's go on to the next page for party party. So we re write that down for part B. We have excellent for O r. And then we have X squared minus X plus one x squared plus two squared. So we will not need long division here because the numerator has degree for where's denominator It will have degree six. So this will be degree for when you square it and then adding two more exes From this leading term, we'LL give you extra the six So no long division. So now we're ready to go straight to the partial fraction the composition. But since these air quadratic ce and the practices, we have to shut off the factor So let's look at X squared plus two. So if polynomial looks like this x x squared plus B X plus c, we need to look at B squared minus four a. C. If this is less than zero, then there's not a factor. The contract is not a factor. So for this polynomial X squared plus two, we have a close one. B zero c equals two and we see that b squared minus four a. C we'LL just be zero minus four times one times two so minus a which is less than zero. So this was not a factor, So that's a quadratic that doesn't factor. And then we could also look at this other quadratic over here. So this one, we'd raise that we have X squared minus X plus one. So for this one, we see a equals one B is minus one. She is one. So these where minus four A. C. We have one minus four times one times one. So we have minus three, which is also negative. So that means that this other quadratic on the bottom left also just impacted. So this is will be what the book calls. He's three. We have this think quadratic factors that irreducible that cannot be factored into smaller parts. So this weaken right is X plus me over X squared minus X plus one. So that's for Case three. And in this term, actually we'LL be using case for for the second term because this one's to the second power. So for this one, because it's a quadratic on the inside we'LL have a CX plus de over X Square plus two a sense It's a repeated factor We'LL have to do this once war but this time on the X squared Plus for two we'LL have to square the very last one and that will be our partial fraction to composition for Barbie

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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.

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27:53

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