Show that for every m and n value, $\int_0^1 x^m (1-x)^n \,dx= \int_0^1 x^n (1-x)^m \,dx$

Question

Show that for every m and n value, $$\int_0^1 x^m (1-x)^n \,dx= \int_0^1 x^n (1-x)^m \,dx$$

I have no idea how to solve a question like that. Do I have to solve both parts of the equation and show that they're equal, or is there any other aspect that I'm not aware of?

Use https://math.stackexchange.com/questions/439851/evaluate-the-integral-int-frac-pi2-0-frac-sin3x-sin3x-cos3x/439856#439856 — lab bhattacharjee, May 28 '20 at 11:51

score 1 · Accepted Answer · answered May 28 '20 at 12:04

1

$$\int_0^1 x^m (1-x)^n \,dx= \int_0^1 x^n (1-x)^m \,dx$$

Your answer is in the image below:

enter image description here

Finally, replace $ t$ by $x$ to get the required answer.

answered May 28 '20 at 12:04

shsh23

1,135

2

Please use LaTeX. Handwritten solutions are not searchable and also run the risk of the images getting corrupted, moved, replaced, or outright deleted. – Cameron Williams May 28 '20 at 12:07
Thank you for the answer! – otnemem May 28 '20 at 12:49

Show that for every m and n value, $\int_0^1 x^m (1-x)^n \,dx= \int_0^1 x^n (1-x)^m \,dx$

1 Answers1