Solving $\int_0^1 [x^{700}(1-x)^{300} - x^{300}(1-x)^{700}] \, dx$ I am trying to solve the following integral:
$$\int_0^1 [x^{700}(1-x)^{300} - x^{300}(1-x)^{700}] \, dx$$
My intuition is that this integral is equal to zero but I am unsure as to which direction to take to prove this. I was thinking binomal expansion but I believe there must be a better way, possibly using summation notation instead.

Well, solving a more general case we have:
$$\mathcal{I}_\beta\left(\text{n},\text{k}\right):=\int_0^\beta\left(x^\text{n}\left(\beta-x\right)^\text{k}-x^\text{k}\left(\beta-x\right)^\text{n}\right)\space\text{d}x\tag1$$
Let $\text{u}=\beta-x$, so we get $-\text{du}=\text{d}x$, so:
$$\mathcal{I}_\beta\left(\text{n},\text{k}\right)=\int_\beta^0-\left(\left(\beta-\text{u}\right)^\text{n}\text{u}^\text{k}-\left(\beta-\text{u}\right)^\text{k}\text{u}^\text{n}\right)\space\text{du}=$$
$$\int_0^\beta\left(\text{u}^\text{k}\left(\beta-\text{u}\right)^\text{n}-\text{u}^\text{n}\left(\beta-\text{u}\right)^\text{k}\right)\space\text{du}=$$
$$\int_0^\beta\left(x^\text{k}\left(\beta-x\right)^\text{n}-x^\text{n}\left(\beta-x\right)^\text{k}\right)\space\text{d}x\tag2$$
So, we get:
$$\mathcal{I}_\beta\left(\text{n},\text{k}\right)+\mathcal{I}_\beta\left(\text{n},\text{k}\right)=0\tag3$$
