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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{4 + x}{(1 + 2x)(3 - x)} $ (b) $ \dfrac{1 - x}{x^3 + x^4} $

(A) $$

\frac{4+x}{(1+2 x)(3-x)}=\frac{A}{1+2 x}+\frac{B}{3-x}

$$

(b) $$

\frac{1-x}{x^{3}+x^{4}}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x^{3}}+\frac{D}{1+x}

$$

Integration Techniques

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University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Okay, So for this problem, we are writing out the form of the partial fraction decomposition without determining the numerical values of the coefficients. So here you have for part a four plus X over one plus two x time stream minus X. So we need thio factor the denominator. And since it's already a nice factor, for we can go ahead and write it as a over one plus two x plus be over three minus X and for part B, we have one minus X over execute plus X for, but it's not in complete. The denominator is not completely factored, so we go ahead and factor out and execute from the bottom. As you can see in my excellent writing you have we get one minus X over execute times the quantity one plus x. And since we have X rays to the third power, we need to take that into consideration and in partial fraction to composition. We write that as follows a over X plus be over X squared plus C over X cubed and then finally plus D over one plus X. And that is Thea Answer. Fix