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(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 $.

(a) $$

\frac{313 x+251}{363\left(2 x^{2}+1\right)^{2}}+\frac{2(2843 x+816)}{3993\left(2 x^{2}+1\right)}-\frac{59096}{19965(5 x-2)}+\frac{5828}{1815(5 x-2)^{2}}

$$

(b) $$

\int \frac{1}{x^{4}+1} d x=\frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{1}{\sqrt{2}} \times\left(x-\frac{1}{x}\right)\right)+\frac{1}{4 \sqrt{2}} \times \ln \left(\frac{\sqrt{2} x+x^{2}+1}{\sqrt{2} x-x^{2}-1}\right)

$$

(c) The graph of the integral is decreasing in intervals: $(-\infty,-0.778)$ and

$(0.803,1) .$

$f(x)$ is undefined at $x=\frac{2}{5},$ so the integral may also have a vertical asymptote

there.

The graph of the integral is increasing in intervals: $\left(-0.778, \frac{2}{5}\right),\left(\frac{2}{5}, 0.803\right)$

asymptote at both ends.

Integration Techniques

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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.