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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} $

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

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

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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.

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Write out the form of the …

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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

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