How Can An Integration Result In Different Expressions?

There is more than one way to integrate  
\[(x-1)^2\]
,
You could expand the brackets and integrate term by term.
\[\int x^2-2x+1 dx=\frac{x^3}{3}-x^2+x+c\]

or you could use the substitution  
\[u=x-1\]
  (then  
\[du=dx\]
) to get  
\[\int u^2 du=\frac{u^3}{3}+c=\frac{(x-1)^3}{3}+c\]

Obviously both expressions are not the same. How can they both be correct?
The arbitrary constants  
\[c\]
  are not the same!
If we expand the brackets for the second answer we get  
\[\frac{(x-1)^3}{3}+c=\frac{x^3-3x^2+3x-1}{3}+c=\frac{x^3}{3}-x^2+ x- \frac{1}{3}+c\]
.
Now both expressions are the same except for the constant terms.
In fact we can write
\[\int x^2+-2x+1 dx=\frac{x^3}{3}-x^2+x+c_1\]

\[\int u^2 du=\frac{u^3}{3}+c=\frac{(x-1)^3}{3}+c_2\]

Where  
\[c_1=c_2- \frac{1}{3}\]
.

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