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 one year ago
Evaluate the integral. The integral will be entered below. The title of this section is Intergration of Rational Fractions by Partial Fractions
 one year ago
Evaluate the integral. The integral will be entered below. The title of this section is Intergration of Rational Fractions by Partial Fractions

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whatisthequestion
 one year ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{}^{}\frac{ x^2x+6 }{ x^3+3x }dx\]

whatisthequestion
 one year ago
Best ResponseYou've already chosen the best response.0@zepdrix does this one only simplify down to \[\int\limits_{}^{}\frac{ x^2x+6 }{ x(x^2+3) }dx\] or am I making a silly mistake somewhere again. If that is as far as it factors down does the (x^2+3) change anything?

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1\[\large \frac{x^2x+6}{x(x^2+3}\qquad = \qquad \frac{A}{x}+\color{royalblue}{\frac{Bx+C}{x^2+3}}\] With partial fractions, you're top part will be one degree less than the bottom. See how the first fraction is a constant over x^1? Our second fraction will be `linear` on top, since the bottom is a quadratic.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1You `could` factor the \(\large x^2+3\) into complex conjugates. But I think that would get kinda messy :) Definitely not worth the effort.

whatisthequestion
 one year ago
Best ResponseYou've already chosen the best response.0okay so then its \[x^2x+6=A(x^2+3)+Bx(x)+C(x)\]

whatisthequestion
 one year ago
Best ResponseYou've already chosen the best response.0could you quickly remind me what a complex conjugate is though? It sounds familiar but I don't know for sure if I have used them before.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1The difference of squares can be broken into conjugates right? \(\large x^2a^2 \qquad = \qquad (xa)(x+a)\) The `sum` of squares produces complex conjgates. It might become relevant if you take differential equations, or complex analysis (I dunno, I haven't gotten that far yet myself). \(\large x^2+a^2 \qquad = \qquad (x+ai)(xai)\)

whatisthequestion
 one year ago
Best ResponseYou've already chosen the best response.0Okay thats the high level stuff my professor randomly talked about the other day then said we didn't need to know about. I knew it sounded familiar.
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