Eli5 how do we know that the integration of x^n is x^(n+1)/n+1

Question

Eli5 how do we know that the integration of x^n is x^(n+1)/n+1

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Answer 1

A simple way it goes the other way

Let m=n+1, it makes the text simple to read.

(x^m)´ = m x^m .

It is well known and you can find the proof at [https://en.wikipedia.org/wiki/Power_rule](https://en.wikipedia.org/wiki/Power_rule)

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We like to remove the initial m in m x^m Multiplying with a constant has not effect (mx)´ =m (x´) so 1/m * (x^m)´ = 1/m * m x^m = x^m

We can now replace m with n+1 and we have (((x^(n+1))/(n+1)) ´ = x^(n+1)

An integral is the reverse of derivation so we get that the integral of x^(n+1) = ((x^(n+1))/(n+1) + C

C it the constant that is often forgotten we you integrate. The derivate of a constant is zero C`= 0 so w could have added that it the derivation calculation too