Question #6a649

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Answer 1 · 2016-09-22T11:34:56

See below.

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

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

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