# Integral - Algebraic Manipulation?

## Homework Statement

$$\int \frac{x}{x+d}dx$$

## The Attempt at a Solution

I tried parts and that turned out horribly so I fired up maple. First move it makes is 'Rewrite Rule'. I've never heard of a rewrite rule but I supposed it could be some algebraic massaging. I ended up looking at it this way:

$$\frac{x}{x+d} = 1-1+\frac{x}{x+d} = \frac{x+d}{x+d}-\frac{x+d}{x+d}+\frac{x}{x+d} = \frac{x+d}{x+d}+\frac{x-d-x}{x+d} = 1-\frac{d}{x+d}$$

Which then is easily integrable. My question is, is there a snappier way to think of this/do it?

Defennder
Homework Helper
Just divide x by x+d and you'll get the term on the rightmost of your equalities.

As in polynomial long division?

edit: Or is there some way that is easy to see what will happen? Basically it isn't straight up intuitive for me to realize that x/x+d is 1-d/x+d, I'm trying to find a quick method to deal with it.

Last edited:
$$\int\frac{x+d-d}{x+d}dx$$

Break it and integrate.

That's much shorter than what I did, thanks.