# Question 6a649

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

$\int {x}^{x} \mathrm{dx}$ 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

${\left(1 + x\right)}^{1 + x} = 1 + x \left(1 + x\right) + \frac{1}{2} {x}^{3} \left(1 + x\right) + \frac{1}{6} \left(x - 1\right) {x}^{4} \left(1 + x\right) + \cdots$

Convergent for $- 1 \le x \le 1$

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#