Integration by Partial Fractions

[burgerandcheese]

Hi. In the second picture (slideshow) it says the coefficients r and s have to be the same sign. But in my coursebook I don't see the authors mentioning anything about this. So do the r and s have to be the same sign or not? Or do they only have to be the same sign when performing integration by partial fractions?

Thank you in advance!

 

[HallsofIvy]

If p and r have different signs, that is if we have either \(\displaystyle px^2- r\) of \(\displaystyle -px^2+ r= -(px^2-r)\) then we have a difference of squares which can be factored further- \(\displaystyle px^2- r= (\sqrt{p}x- \sqrt{r})(\sqrt{p}x+ \sqrt{r})\). I'll bet your text book, while not saying anything about "signs" did say that form was to be used for quadratic terms that could not be factored into linear terms.

 

[burgerandcheese]

If p and r have different signs, that is if we have either \(\displaystyle px^2- r\) of \(\displaystyle -px^2+ r= -(px^2-r)\) then we have a difference of squares which can be factored further- \(\displaystyle px^2- r= (\sqrt{p}x- \sqrt{r})(\sqrt{p}x+ \sqrt{r})\). I'll bet your text book, while not saying anything about "signs" did say that form was to be used for quadratic terms that could not be factored into linear terms.

Oh I see. Thanks again. And yes it does say the quadratic factor cannot be factorised.