Using partial fractions - why is it Ax+B/x^2 and not A/x + B/x^2?

Thread starter IntegrateMe
Start date Jan 19, 2010
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Fractions Partial Partial fractions

In summary, the form for partial fractions Ax+B/x^2 is used to simplify expressions and make them easier to integrate. The terms A and B cannot be combined with the variable x in the denominator, so the form Ax+B/x^2 cannot be simplified. The values of A and B can be found by equating the numerators of the original fraction and the partial fractions form. When using partial fractions, specific steps must be followed, including factoring the denominator, setting up the partial fractions form, and integrating and combining the terms. Partial fractions can only be used when the denominator can be factored into linear or quadratic terms.

Jan 19, 2010

IntegrateMe

Partial fractions of (-2x²+10x+8)/[x²(x+2)]

I initally thought that it was A/x + Bx+C/x² + D/x+2 but you really just do Ax+B/x² + C/x+2 ...can anyone explain why the "x²" isn't split?

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Jan 19, 2010

Dick

Science Advisor

Homework Helper

A/x+(B+Cx)/x^2=A/x+B/x^2+C*x/x^2=A/x+C/x+B/x^2. Doesn't that make the C term sort of redundant?

FAQ: Using partial fractions - why is it Ax+B/x^2 and not A/x + B/x^2?

Why is the form for partial fractions Ax+B/x^2 and not A/x + B/x^2?

The form for partial fractions Ax+B/x^2 is used because it allows us to simplify the expression and make it easier to integrate. The fraction A/x represents a simple term, while B/x^2 represents a term with a denominator that can be integrated using the power rule. This form also helps us solve for the constants A and B more easily.

Can we simplify the form Ax+B/x^2 to A/x + B/x^2?

No, we cannot simplify the form Ax+B/x^2 to A/x + B/x^2. The terms A and B are constants and cannot be combined with the variable x in the denominator. Therefore, we keep the form Ax+B/x^2 as it is.

How do we determine the values of A and B in the partial fractions form Ax+B/x^2?

The values of A and B can be determined by finding a common denominator and equating the numerators of the original fraction and the partial fractions form. This will create a system of equations that can be solved to find the values of A and B.

Is there a specific method to follow when using partial fractions?

Yes, there are specific steps to follow when using partial fractions. First, you need to factor the denominator of the original fraction into linear or quadratic terms. Then, you need to set up the partial fractions form with undetermined constants. The next step is to equate the numerators and solve for the constants. Finally, you can integrate the partial fractions and combine the terms to get the final solution.

Can we use partial fractions for any type of fraction?

No, we cannot use partial fractions for every type of fraction. Partial fractions are used when the denominator of the original fraction can be factored into linear or quadratic terms. If the denominator has higher-order terms or is irreducible, then partial fractions cannot be used.