- #1

MAGNIBORO

- 106

- 26

I try to solve the integral $$\int_{0}^{1} x^{x} dx$$

Through sums of riemann But I came to the conclusion that the result is 0 that is wrong

$$\int_{0}^{1} x^{x} dx = \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}$$

$$= \lim_{n\rightarrow \infty }\frac{\frac{1}{n}^{\frac{1}{n}}}{n} + \frac{\frac{2}{n}^{\frac{2}{n}}}{n} + ... +\frac{\frac{n-1}{n}^{\frac{n-1}{n}}}{n} + \frac{1}{n}=0$$

$$\int_{0}^{1} x^{x} dx = 0 $$

I'm sure the mistake is expand the ##\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}##

because in some easy integrals like ##\int_{0}^{1} x dx## if you expand the ##\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )## you get

$$= \lim_{n\rightarrow \infty } \frac{1}{n^{2}}+\frac{2}{n^{2}}+...+\frac{n-1}{n^{2}}+\frac{1}{n} = 0$$

Instead of finding a "closed" formula like:

$$= \lim_{n\rightarrow \infty } \frac{n(n+1)}{2\, n^{2}}=\frac{1}{2}$$

so If I can not find a formula like ## \frac{n(n+1)}{2}## for ## \sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}##

the limit of the riemman sum is wrong?