Need help doing simple re-arrangement

[adelaide87]

Homework Statement

How do you go from:

(2x)/(2x+1)

and rearrange to get:

1-[(1)/(2x+1)]?

My book goes directly from 1 to the other before doing the integration and both are the same, but can someone explain how this rearrangement is done?

Thanks!

 

[Doc Al]

You can (almost) always replace 1 by (anything)/(anything). In this case they used 1 = (2x+1)/(2x+1).

 

[njama]

They used little trick:
$$\frac{2x}{2x+1}=\frac{(2x+1)-1}{2x+1}=1-\frac{1}{2x+1}$$

 

[adelaide87]

Sorry haha but I am still confused as to what you did exactly.

 

[VeeEight]

The first step is that 2x = (2x + 1) - 1

$$\frac{(2x+1)-1}{2x+1}= \frac{2x+1}{2x+1} - \frac{1}{2x+1} = 1-\frac{1}{2x+1}$$since $$\frac{2x+1}{2x+1} = 1$$

 

[adelaide87]

so what about this one:(3x)/(4x+1)

I don't understand if you are just bringing the bottom up, or how 2x = (2x + 1) - 1

 

[adelaide87]

Okay nevermind, I see what you are doing. Didnt make sense at first, thank you for the help.