Beta Function

[lucid magnet]

So how do i solve this i only solved exercises with integrals from 0 to 1.
I can see there is a relationship between the upper bound 1/2 and 2x but I can't figure out how to solve it.

[old shell]

Take u = 1 - 2x, x = (1/2)(1 - u), dx = -(1/2)du. Then:
∫(x(1 - 2x)^7 dx, 0, 1/2) = -(1/4)∫((1 - u)u^7 du, 1, 0) = (1/4)∫(u^7 (1 - u)du, 0, 1)
Now just apply the definition of beta function.

And, in this case, you can also easily verify the answer, as the integral is quite simple anyway.

[lucid magnet]

wait so since dx=-1/2du you divided the lower and upper bounds by 1/2 and reversed them because -1/2 is a negative number?

[old shell]

No.
Since u = 1 - 2x, the lower limit becomes 1 - 0 = 0 and the upper limit becomes 1 - 1 = 0.
After that, I swapped them to make their order natural using the minus that we got from dx = -(1/2)du.

[lucid magnet]

thank you i understand now