x-x-dx-

x-x-dx-

Question Number 99403 by I want to learn more last updated on 20/Jun/20

\int x^{x} dx

Commented by PRITHWISH SEN 2 last updated on 20/Jun/20

there is no possible soln . for this integral .

Answered by smridha last updated on 21/Jun/20

\int x^{x} d x = \int e^{x l n (x)} d x = \underset{n = 0}{\sum^{\infty}} \frac{1}{n!} \int x^{n} {(l n [x])}^{n} d x

l e t x = e^{k} s o = \underset{n = 0}{\sum^{\infty}} \frac{1}{n!} \int e^{(n + 1) k} k^{n} d k

= \underset{n = 0}{\sum^{\infty}} \frac{1}{n!} [\underset{m = 0}{\sum^{\infty}} \frac{1}{m!} {(n + 1)}^{m} \int k^{m + n} d k]

= \underset{n, m = 0}{\sum^{\infty}} \frac{1}{n! m!} . \frac{{(n + 1)}^{m}}{(m + n + 1)} . {[l n (x)]}^{m + n + 1} + c

Commented by I want to learn more last updated on 23/Jun/20

Thanks sir