Integration Proof

Proof of $\int x^{n}dx=\frac{x^{n+1}}{(n+1)}$

If

$$\frac{d}{dx}{x}^n=n{x}^{n-1}$$

integrating, we get,

$$x^n=\int n{x}^{n-1}dx$$

Now let

$$n⟼n+1$$

then

$$x^{n+1}=\int (n+1)x^{n+1-1}dx$$ $$x^{n+1}=(n+1)\int x^{n}dx$$ $$\int x^{n}dx=\frac{x^{n+1}}{(n+1)}$$

... as required.