Integrating $\int x+1 \, dx$ using $u$-substition gives wrong answer. $\int (x+1) dx$
Substituting:
$u = x + 1$ and $du = dx$
$$\int u \,du = \frac{u^2}{2}$$
Which gives:
$$\int (x+1) \, dx = \frac{(x+1)^2}{2}$$
This is completely wrong as the correct answer is $\frac{x^2}{2}+x$
Why did this happen?
 A: You forgot the constant C. It is very important:
$$
\int u du = \frac{u^2}{2} + C = \frac{(x+1)^2}{2} + C = \frac{x^2}{2} + x + \frac{1}{2}+ C = \frac{x^2}{2} + x + C'
$$
A: If you expand your second answer, you get
$$ \frac{(x+1)^2}{2} = \frac{x^2}{2} + x + \frac{1}{2}. $$
This differs from the "correct" answer by a constant.  But if two functions differ by a constant, they will have the same derivative.  Hence both answers are equally correct (or incorrect).  I submit that a truly correct answer would be
$$ \frac{x^2}{2} + x + C
\qquad\text{or}\qquad
\frac{(x+1)^2}{2} + C, $$
where $C$ is a real constant.

Addendum:  I have a stupid trick for getting my students to remember the constant.  We have
$$ \int \frac{1}{\text{cabin}} \, \mathrm{d}(\text{cabin}) = \text{house boat}.$$
Why?
Remember that
$$ \int \frac{1}{x}\, \mathrm{d}x = \log|x| + C. $$
Thus
$$ \int \frac{1}{\text{cabin}} \, \mathrm{d}(\text{cabin})
= \log|\text{cabin}| + \text{sea}, $$
which is quite clearly a house boat.
A: Both answers are right: $\displaystyle \frac{(x+1)^2} 2 = \frac{x^2 + 2x + 1} 2 = \frac {x^2} 2 + x + \text{constant}$
