Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

(a) Find the partial fraction decomposition of the function$$ f(x) = \frac{12x^5 - 7x^3 - 13x^2 + 8}{100x^6 - 80x^5 + 116x^4 - 80x^3 + 41x^2 - 20x + 4} $$(b) Use part (a) to find $ \int f(x) dx $ and graph $ f $ and its indefinite integral on the same screen.(c) Use the graph of $ f $ to discover the main features of the graph of $ \int f(x) dx $.

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Campbell University

Harvey Mudd College

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.

12:11

(a) Use a computer algebra…

01:07

Determine the following us…

03:56

Find the partial-fraction …

05:21

Write out the form of the …

So for part A, let's go ahead and find the partial fraction decomposition. So for this problem in the text book, we see that next to the problem, there's a C A s symbol. So this means computer algebra system. So that's what we'll be using here. So in the computer algebra system Wolfram Alpha, I have inserted a fraction that we have now. If I scroll down, I see the partial fraction decomposition. So this is what we'll use for apart A. So I'll go ahead and write this down. So after Becks, so partial fraction given by Wolfram and then the next term. Oh, then I have 816 all over 3993 two X squared, plus one. And then two more terms. Mm, I'm And then finally the last term. So that's the partial fraction decomposition given by Wolfram. Now let's go ahead and use Wolfram and a calculator to find Part A and it's integral. Let's graph those on the same screen, so let me go to the next page here. So for part B, they want the integral of F. So using our computer algebra system. If I just scroll down here am I wolf Front page. We see that there's an indefinite integral that's given to us. So here's our original fraction F of X. Here they give us the anti derivative. So let's just go ahead and write that in for Barbie. So let me. Actually, I'll need a lot of more room here, so let me start down here, all right? It just how Wolfram does. And then we have natural log. You don't have to have absolute value here because two X squared plus one is positive. Yeah. Okay. Yeah. And then we're still inside of this. Large parentheses here. And the next term. Yeah. Then I have natural log. Let me take a step back. Getting a little sloppy here. 84 natural log two minus five x in the absolute value. And then the last term here and then at our constant of integration, see, And we'll close off the parentheses. So this was all multiplied by the fraction outside. So that's our anti derivative. And then for a part of C Well, and actually we're still on part b tool need to graph the original function, then part A and then the integral part B So here and here's part a of top original function that we had, and then on the bottom, I just inserted in blue, the anti derivative. And so if we look at the graph, we can try to convince ourselves that the red graph is the original and that the blue graph is the anti derivative. So here, for example, if this is true, this means that the red graphs should be the tangent line of the blue ref, and we can basically see that's true. Whenever the blue graph is increasing, the red graph is positive. Over here, blue graph is increasing. Red graph is so positive and so on. So this is just more evidence that we've graphed the correct functions, and that's our final answer.

View More Answers From This Book

Find Another Textbook

Missouri State University

Baylor University

University of Michigan - Ann Arbor