Integrating $\int (x+1)^2 dx$ two ways gives different results: $\frac13 x^3+x^2+x$ vs $\frac13 x^3+x^2+x+\frac13$. Why? I was trying to compute $$\int (x+1)^2 \, dx~,$$ which is a really easy function to integrate. 
But the thing is that I write the function as $x^2 +2x+1$ and the result I got was $\frac{x^3}{3} +x^2 +x$. 
But then I tried to integrate it using substitution; I called $u=x+1$, so the function I had to integrate was $$\int (u)^2 \, du~,$$ and the result is $\frac{u^3}{3}$ and then $\frac{(x+1)^3}{3}$, which is equal to $\frac{x^3}{3} +x^2 +x+\frac{1}{3}$ and that's different from the previous result!

WARNING while dealing with INDEFINITE INTEGRALS!!
When you solve some indefinite indefinite integral you should always add a constant in your final result (why?). Remember that indefinite integral is nothing but anti-derivative. Therefore, whenever you integrate you miss a constant which disappears because the derivative of constant is $0$.
