Question #6a649

1 Answer
Sep 22, 2016

See below.

Explanation:

This is an integral that cannot be obtained in closed form. In those cases you can try to integrate the series representation to obtain an answer.

Taking as an example

#int x^x dx# which is also an integral not obtainable in closed form,
developping in series

#(1+x)^(1+x) = sum_(k=0)^oo((Pi_(j=1)^k(x-j+2))/(k!))x^k# or

#(1+x)^(1+x) = 1 + x (1 + x) + 1/2 x^3 (1 + x) + 1/6 (x-1) x^4 (1 + x)+cdots#

Convergent for #-1le x le 1#

Now, with #0 le x le 2# you can approximate the integral obtaining

#intx^xdx approx sum_(k=0)^nint((Pi_(j=1)^k(x-j+1))/(k!))(x-1)^kdx#