Answer
See explanation.
Work Step by Step
Let $d u=-d x$ and $-x+1=u$
Simplify
$\int_{0}^{1} x^{m}(-x+1)^{n} d x=-\int_{1}^{0}(-u+1)^{m} u^{n} d u$
Rewrite integral
$\int_{0}^{1} x^{m}(-x+1)^{n} d x=\int_{0}^{1}(-u+1)^{m} u^{n} d u$
Replace X with U, du with dx
$\int_{0}^{1}(-x+1)^{m} x^{n} d x=\int_{0}^{1} x^{m}(-x+1)^{n} d x$