How Can An Integration Result In Different Expressions?

There is more than one way to integrate  
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  
) 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  
  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
\[\int x^2+-2x+1 dx=\frac{x^3}{3}-x^2+x+c_1\]

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

\[c_1=c_2- \frac{1}{3}\]

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