When studying various functions and proofs, I've often seen the following...
.
Which is the reason why some functions such as the Beta function can be written as...
$$B(x,y) = B(y,x)$$
What I want to know is why this proof holds true? What is a good explanation for someone that is relatively new to integrals? I would like to be able to prove this to myself and visualize it better.
Looking at it alone doesn't make sense to me, but for some reason the rule holds.
Starting with $$\int_{0}^af(t)dt, $$ consider the substitution $t = a-u.$ Then $dt = -du$ and the limits $t=0$ and $t=a$ become the limits $u = a$ and $u = 0$. Hence $$\int_{0}^af(t)dt = \int_{a}^0f(a-u)(-du) = -\int_{a}^0f(a-u)du = \int_{0}^af(a-u)du.$$ Since $u$ is a dummy variable in this last step, we may simply change each $u$ to a $t$ and we end up with the form you have.