Indefinite Integrals - which method is preferred?

[1MileCrash]

Homework Statement

$\int (x+1)^2 dx$

Homework Equations

The Attempt at a Solution

I am just getting into this, and this is a simple problem, but my book and I took two separate routes. My question, essentially, is does any constant you get just "combine" with the "any constant" C?

I went with:

$\int (x+1)^2 dx = \int x^2+2x+1 dx$

Which yields
$\frac{1}{3}x^3+x^2+x+C$

Now, my textbook took a wildly different method, with u substitution and arrived at:
$\frac{(x+1)^3}{3}+C$

These equations are not identical, but their derivatives are. So, they are both solutions and they are both the same. My gripe with the textbook solution is that, their answer can be brought to mine but with another constant. Why would they do that?

$\frac{(x+1)^3}{3}+C$

$=\frac{x^3+x^2+3x+1}{3}+C$

$\frac{1}{3}x^3+x^2+x + \frac{1}{3} + C$

Why would they give that as a final result? Their answer is just mine with another constant hidden in. Is there are reason for them to do that?

 

[Dick]

Their method can easily solve the integral of (x+1)^100 as (x+1)^101/101+C. There's really no profit in expanding the polynomial. Yours would do it too, but it would be painful. That's why they are doing it that way.

 

[1MileCrash]

Their method can easily solve the integral of (x+1)^100. Yours would do it too, but it would be painful. That's why they are doing it that way.

I would use their method in that case, though.

Or are you suggesting that they just did it that way to show the process?

 

[Dick]

I would use their method in that case, though.

Or are you suggesting that they just did it that way to show the process?

If they expanded the final polynomial, which they didn't need to do, then it was probably just to make it clear that the results are the same as the way you did it, just with a modified constant. If you just write the answer as (x+1)^3/3+C, that's already easier than expanding and integrating term by term, isn't it? In general, I prefer to do things the easiest way, even though I know harder ways will work as well.