Integrate (5x+2)dx/(x-2) from 0 to 1

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SUMMARY

The integral of \(\frac{(5x+2)dx}{x-2}\) from 0 to 1 can be simplified using the substitution \(u = x - 2\), which leads to \(dx = du\) and \(x = 2 + u\). An alternative method involves rewriting the numerator as \(5(x-2) + 12\), allowing for polynomial division to yield \(5 + \frac{12}{x - 2}\). This approach clarifies the substitution process and simplifies the integration task. The discussion emphasizes the importance of recognizing algebraic manipulation techniques in solving integrals.

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Homework Statement



\int\frac{(5x+2)dx}{x-2} from 0 to 1


Homework Equations



The Attempt at a Solution



I've tried splitting it up into (5x)/(x-2) + (2)/x-2), but i couldn't go any farter. Ived also tried using lots of U subsitutions, but i can't figure out what do next. Is there some trick that i am not seeing?
 
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Yep, the obvious one. u=x-2. dx=du. x=2+u.
 
Another way to split it up [without explicitly invoking a substitution]
is to write the numerator 5x+2 as 5(x-2)+12.
 
robphy said:
Another way to split it up [without explicitly invoking a substitution]
is to write the numerator 5x+2 as 5(x-2)+12.
Woah, woulda never thought of that. Nice, I want your vision :-]
 
robphy said:
Another way to split it up [without explicitly invoking a substitution]
is to write the numerator 5x+2 as 5(x-2)+12.

You'll still want u=x-2 to do the 12/(x-2) part.
 
Just do polynomial division. It becomes 5 + 12/(x - 2).

Oops.
 
Dick said:
You'll still want u=x-2 to do the 12/(x-2) part.

True... and now the substitution is really obvious.
 

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