Let $$I = \int_{a}^{b}(x - k)^2 dx$$ then
$$I = \int_{a}^{b}(x^2 - 2xk + k^2) dx = \left[\frac{x^3}{3} - x^2k + xk^2\right]_a^b$$
but also if we let $u = x - k \Rightarrow du = dx$ with $x = a \Rightarrow u = a - k, x = b \Rightarrow u = b - k$ then
$$I = \int_{a - k}^{b - k} u^2 du = \left[\frac{u^3}{3}\right]_{a - k}^{b - k} = \left[\frac{(x - k)^3}{3}\right]_{a}^{b} = \left[\frac{x^3}{3} - x^2k + xk^2 - \frac{k^3}{3}\right]_a^b$$
Which obviously can't be true unless $k = 0$. What am I doing wrong here?
$k$ is a constant with no terms of $x$ associated with it. So on substituting the limits,
$-\frac{k^3}{3}$ will get cancelled by $\frac{k^3}{3}$.