TITLE: Solve $\int_0^1((1-x)^8x^{11}-(1-x)^{11}x^8)dx$
QUESTION [5 upvotes]: 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

REPLY [3 votes]: Since people are giving alternate approaches to the $t = 1 - x$ substitution, here's another one:
Integrate by parts in ${\displaystyle \int_0^1 (1 - x)^8x^{11}\,dx}$, integrating the $(1 - x)^8$ and differentiating the $x^{11}$. The endpoint terms are zero, so we get
$$\int_0^1 (1 - x)^8x^{11}\,dx = {11 \over 9} \int_0^1 (1 - x)^9x^{10}\,dx$$
Doing the same thing two more times yields
$$\int_0^1 (1 - x)^8x^{11}\,dx = {11 *10 * 9 \over 9 * 10 * 11} \int_0^1 (1 - x)^{11}x^8\,dx$$
This can be rearranged as 
$$\int_0^1 \big((1 - x)^8x^{11} - (1 - x)^{11}x^8\big)\,dx = 0$$