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If $ a \neq 0 $ and $ n $ is a positive integer, find the partial fraction decomposition of$$ f(x) = \frac{1}{x^n (x - a)} $$[Hint: First find the coefficient of $ \frac{1}{(x - a)} $. Then subtract the resulting term and simplify what is left.]

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Oregon State University

University of Michigan - Ann Arbor

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.

02:18

If $a \neq 0$ and $n$ is a…

00:44

Determine the partial frac…

03:22

Partial Fraction Decomposi…

03:15

00:15

Find the partial-fraction …

02:27

Writing the Partial Fracti…

00:43

01:10

Find the partial fraction …

Let's find the partial fraction in opposition. And here let's not that we're given is non zero. So here we can rewrite this. So for the first term X event, let's use what the author calls case, too. This is when you have repeated linear factors. So then a two over X squared and we go all the way up xn and then we'LL have a case One. This is a non repeatedly in your factor X minus a with good and multiply And both sides by the denominator on the left So on the right hand side and so on and we'LL have all the way up to a and X minus a and then we'LL have be accident Now the next step would be to go ahead and combine depending on the power of X So flu pullout xn one a one plus me Pull out a xn minus one. You get a two minus a one and so on. And when we plotted X day and minus a a and minus one and then finally the constant term and a So now we have comparing the constant terms We have a one on the left. We have a negative and and the right So that gives us a N equals negative one over, eh? And for instance, we can use that to find this. We know this is equal to zero because there's no X on the left. So we have a end here, Let me write it this way. And minus one equals a A and over, eh equals negative. A one over a square. Similarly, a N minus two equals a and minus one. Over, eh? So here, negative one. Over a cute. That's a negative up there and continuing in this fashion level. Me Go on to the next. Actually, let me take a step back here. That should have been in minus one. So what I had. Okay, so let me go. The next page, a one is a two over, eh? A three over a square. Just using the same formula repeatedly. Negative one over eight, then and then be equals negative. A one, because a one plus zero zero zero and then this is positive. One over. And so now that we've found all of our coefficients for the partial fraction to composition, just plug a moment. Negative one. An ex those are capital that was a one and then a two and minus one X squared and so on. Great. And then finally negative. One of rain and exit and and then R B was won over eight of the end, and that's her room was X minus a. There it is. That's our final answer.

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