x-x-dx-

x-x-dx-

Question Number 62906 by Prithwish sen last updated on 26/Jun/19

\int x^{x} dx

Commented by mathmax by abdo last updated on 27/Jun/19

l e t A = \int x^{x} d x \Rightarrow A = \int e^{x l n (x)} d x = \int \sum_{n = 0}^{\infty} \frac{x^{n} {(l n (x))}^{n}}{n!} d x

= \sum_{n = 0}^{\infty} \frac{1}{n!} \int x^{n} {(l n (x))}^{n} d x = \sum_{n = 0}^{\infty} \frac{W_{n, n}}{n!} w i t h W_{n, p} = \int x^{n} {(l n x)}^{p} d x

b y p s r t s W_{n, p} = \frac{1}{n + 1} x^{n + 1} {(l n (x))}^{p} - \int \frac{1}{n + 1} x^{n + 1} \frac{p}{x} {(l n (x))}^{p - 1} d x

= \frac{1}{n + 1} x^{n + 1} {(l n (x))}^{p} - \frac{p}{n + 1} \int x^{n} {(l n x)}^{p - 1} d x = \frac{1}{n + 1} x^{n + 1} {(l n x)}^{p} - \frac{p}{n + 1} W_{n, p - 1}

\Rightarrow W_{n, n} = \frac{1}{n + 1} x^{n + 1} {(l n x)}^{n} - \frac{n}{n + 1} W_{n, n - 1} \Rightarrow

A = \sum_{n = 0}^{\infty} \frac{{(l n x)}^{n}}{n + 1} x^{n} - \sum_{n = 1}^{\infty} \frac{n}{n + 1} W_{n} \dots b e c o n t i n u e d \dots .

Commented by Prithwish sen last updated on 27/Jun/19

thank you sir

Commented by mathmax by abdo last updated on 27/Jun/19

y o u a r e w e l c o m e s i r .