I'm trying to evaluate a simple integral with basic rules we learned : $$\int\frac{2t+3}{9t^2-12t+8}dt$$ However I try, I fail. I tried substitution, splitting into two integrals and also square completion so I have this : $$\int\frac{2t+3}{9(t-\frac{2}{3})^2+4}dt$$ But it still leads me nowhere. What am I doing wrong?
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Hint. You can split the given integral into $$ \int\frac{2t+3}{9t^2-12t+8}dt=a\int\frac{(9t^2-12t+8)'}{9t^2-12t+8}dt+b\int\frac{1}{9(t-\frac{2}{3})^2+4}dt. $$ Can you finish it?
Olivier Oloa
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You find that $a=\dfrac19$, $b=\dfrac{13}3$. – Olivier Oloa Mar 18 '17 at 23:02
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Thank you for the answer. I can figure a, but not b. I can't see how this fraction is helping.. – shlogek Mar 18 '17 at 23:06
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You want to have $18at-12a+b=2t+3$ for all $t$, this gives $18a=2$ and $-12a+b=3$, thus $b=3+12a=3+12/9=13/3$. – Olivier Oloa Mar 18 '17 at 23:09
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I get it now, thank you very much for your time. – shlogek Mar 18 '17 at 23:11
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@shlogek You are very welcome. – Olivier Oloa Mar 18 '17 at 23:14
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Symbolab may help you
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It might be better to explain each step taken toward evaluating the integral rather than post a link to an algorithm. – Santana Afton Mar 18 '17 at 23:14
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I mean im not a professor or anything, the guy was asking for help, and I found the answer on Symbolab, why not share? – Ricardo Silvestre Mar 18 '17 at 23:18
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@RicardoSilvestre To answer your question "why not share", please see the help page for mathstackexchange.com: http://math.stackexchange.com/help/how-to-answer, the section about sharing links. – Χpẘ Mar 19 '17 at 00:12