When should I use "partial fraction decomposition" when integrating? [closed]

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Does the difference in power have to be greater than 1? How do I know when to use u-substitution and when to use long division and when to use decomposition etc.?

Thanks

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1 Answer

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When integrating a rational function, i.e. the quotient of two polynomials, you use long-division if the degree of the top polynomial isn't smaller than that of the bottom one. This allows you to obtain a proper rational function - one in which the bottom polynomial is of higher degree - that you can then integrate on.

When integrating a proper rational function, you can re-express the fraction using partial fraction decomposition. If you can decompose the bottom polynomial, and hence can write the partial fraction decomposition of your proper rational function, then you can solve the integral easily.

So you should use the partial fraction decomposition when integrating a proper rational function - unless the function is already made up of simple fractions that you can integrate directly (such as $A/(x-c)^n$, or $(Ax+B)/(x^2+bx+c)$ where $x^2+bx+c$ has no roots). Or unless you don't know how to decompose the bottom polynomial into binomials and trinomials, so you can't write down the partial fraction decomposition.

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