Integration Question: Need Explanation in English

[neshepard]

Homework Statement

∫2x/(2x+1)dx

Homework Equations

The Attempt at a Solution

I know from the web the answer is -x+1/2ln(2x+1)+C, but all I keep seeing for explanations is add and subtract 1 to the numerator and mysteriously you get an answer. I don't get it. Can somebody Please, Please, Please, explain, or point me to a site that helps to explain how to integrate this in detail? I just need something in english, not math language.

Thanks

 

[rock.freak667]

The principle they are applying is that 1-1 = 0

so if I had just 'x' and I write back 'x+1-1', I'd still have 'x' right? It is the same with your example.


$$\int \frac{2x}{2x+1} dx = \int \frac{2x+1-1}{2x+1} dx$$

Which can then split since we know hat (a+b)/c = a/c + b/c

$$\int \frac{2x+1-1}{2x+1} dx = \int \left( \frac{2x+1}{2x+1}- \frac{1}{2x+1} \right) dx$$

Is it a bit clearer now?

 

[neshepard]

That's what I keep seeing. And I do get your x+1-1=x, but we never did a problem exactly like this in calc 1 and therefore I don't get why you decided to do that. Is it always the case where I do the x+1-1=x thing? So for 3x/(3x+4) I would do
3x+4-1/(3x+4) and finish the problem las above?

I know it sounds like I'm an idiot, but I'm the type that needs to foundation of "why" and then I can solve the math.

Thanks for the help.
Neil

 

[stevenb]

I know it sounds like I'm an idiot, but I'm the type that needs to foundation of "why" and then I can solve the math.

No, you're not an idiot, but you maybe need to get a little more into the mathematical spirit of things.

Integration is one of those tasks that often requires some nice tricks, like the one you're asking about now. Over time, you start to collect a bag of more and more tricks as you learn them from others, or figure them out yourself. This is part of the fun of it. Sometimes you see a trick right away, while other times it takes days of thinking with the final solution waking you up in the middle of the night.

How did you do with solving trig identities back in trigonometry class? That's another task that takes some insight and experience. What about doing geometric proofs in geometery class? Or proofs in general?

All of these things come more easily to some people, but anyone can improve their skill with experience.

 

[rock.freak667]

Well essentially that technique saves time instead doing polynomial division.

 

[Bohrok]

Is it always the case where I do the x+1-1=x thing? So for 3x/(3x+4) I would do
3x+4-1/(3x+4) and finish the problem las above?

For 3x/(3x + 4) you want to write the numerator as 3x + 4 - 4, not 3x + 4 -1 because this isn't equal to 3x. You want 4 - 4 because that's basically 0; adding 0 can be very useful for some problems such as these.

 

[neshepard]

Thanks to all. Bohrok, I meant to type 4-4 and not 4-1. I've spent 2 hours trying to figure this problem and the next in my homework and have a splitting headache which caused the 4-1 typo, but thanks for the help.