If $ a $ and $ b $ are positive numbers, show that $$ \int^1_0 x^a(1 - x)^b \,dx = \int^1_0 x^b(1 - x)^a \,dx $$
Hello. The caution is taken from calculus and the ocean is if A. And B are positive numbers show that integration 021 X. To the power of a 1 -1. to the power will be the F. X. Is equal to integration. Acts to the power bi and one minus X to the power A. Dfx. Okay. We need to evaluate, we need to find, we need to show that both of these are equally. So let me evaluate it. So let me take the left hand side which is equal to integration 0 to 1 extra. The power A. And one minus X. To the power bi into dx. Let when minus X is equal to the then minus D. X. Is equal to duty. So and the limit will be when X is equal to one Mhm T. Is equal to View and when X is equal to zero T. Is equal to one. So let me substitute these values in the equation we get integration 1 to 0 X. To the power A. Become 1- T. to the power a. And when minus X become T. To the power will be and the excess minus of DT. We know that 02 for any equation. 02 X F f X D X. Can be written as miners of X. 20 actual facts. The X. I'm using that same condition in the equation. So we get That is equal 0-1, one minus the to the power A. And T. To the power bi into DT. He's just a dummy variable. So we can replace tv X. And finally we get the condition The Old 1, 1 -X. To the power A. And X to the power bi. Into the X. It's the same as our right hand side. So just replace, we can be able to one X. To the power bi. And one minus X. To the power bi fx the same as a right hand side. So hope this clears your garden.