I have a question related to definite integrals and series from this site here.
It is written that the definite integral of $\max(x,1-x)dx$ from $0$ to $1$ is equal to $\frac34$:
$$ \int_0^1 \max (x, 1-x) dx = \frac34$$
but I have question, there are two different cases (I don't consider when $x$ is between $0$ and $1$, because in this case it is undefined which one is maximum), but in the second case, if $x<0$, then it is clear that $1-x$ is greater than $x$, so its integral is $x-x^2/2$, and after plugging values,we get $1/2$, and on the other hand, if $x>1$, then $x$ is maximum, its antiderivative is $x^2$ so we get $1/2$, so when is it equal to $3/4$? Please help me to understand this problem.