Solve $\int_0^1((1-x)^8x^{11}-(1-x)^{11}x^8)dx$ I have got to solve this definite integral, though I have no idea in which direction to go. Another exercise I had is to solve something similar that looks like:
$$\int_0^1((1-x)^{11}x^2)dx$$
so for this integral I could substitute $t=1-x$ and I get a much easier version of the integral. The same method I couldn't implement on the given integral $\int_0^1((1-x)^8x^{11}-(1-x)^{11}x^8)dx$ as it just doesn't work.
The only thing I could notice is:
$$\int_0^1((1-x)^8x^{11}-(1-x)^{11}x^8)dx=\int_0^1((1-x)^8x^8(x^3-(1-x)^3))dx$$
yet I'm stuck at here and I have no clue how to solve the integral. (sure there is the simplest solution just to open up the polynomial - but it's naive and foolish...)
thanks

The symmetry of the expressions suggests shifting to center the integral around $0$. Substituting $u=x-1/2$, you have
$$\int_{-1/2}^{1/2}\left(\left(\frac12-u\right)^8\left(\frac12+u\right)^{11}-\left(\frac12-u\right)^{11}\left(\frac12+u\right)^8\right)du$$
The integral is of an odd function over $[-a,a]$, and so its value is $0$.
