Help in an integration proof

[teal lava]

So I'm a high school student studying high school and college level calculus at once and I have been able to solve every integral problem I've come across up until I've come across this problem in Hebrew: prove that if m and n are natural numbers then the following equation holds true, tip: don't calculate, just integrate by substitution.

[true coral]

**Hint: ** make the change of variable , ||t=1-x||

[deft lintel]

trigonometria
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[true coral]

This is just for the laughs, I recommend using the first approach

[teal lava]

I tried substituting t for 1-x I still get stuck with the problem of exponentiation because of the constants m and n

[true coral]

$$\int_0^1x^m(1-x)^n \space dx\stackrel{u=1-x}{=}\int_1^0(1-u)^mu^n(-1)du$$ $$=\int_0^1(1-u)^mu^ndu=\int_0^1x^n(1-x)^mdx$$

[deft lintel]

trigonometria

[pseudo flax]

directly use kings property

[deft lintel]

𝐃𝐞𝐚𝐝 𝐊𝐢𝐥𝐥𝐞𝐫 𝟎𝟒

[pseudo flax]

here a=0, b=1, so proven by identity

[true coral]

I'd argue that you cannot prove king's rule using king's rule.

[pseudo flax]

actually, its an identity

and it can be proven by symmetricity of graph

[teal lava]

Something seems missing

I can see where you're going with the first three equations

But the way you connected the fourth one seems unclear

You might've substituted something incorrectly

[true coral]

$\int_a^b f(x) dx=\int_a^b f(u)du$

[deft lintel]

trigonometria

[true coral]

x is just a variable so we can change it as we wish

[teal lava]

Yes but how are the first three equations, of course all equal to one another, equal to the fourth one

As long as u=1-x the equation remains unchanged

[true coral]

i just changed u for** x** .

[teal lava]

I used king's property and it turned out to be a really reliable method

[teal lava]

It's possible to change the value of a substitution variable?

[true coral]

yes it is just a dummy variable no real "information" is stored

[teal lava]

Thanks so much

So note to self: we can change u from u=1-x to u=x

[true coral]

that is only beacuse it is a definite integral though.

because the limits of integration change according to our variable any time we substitute

[pseudo flax]

ye it is

[true coral]

so letting x=u at the end does nothing

[teal lava]

Thank you guys so much