Integral Calculus

integrate, ∫ X^X dx

madari bhimji kamji

11 Years agoGrade

1 Answer

Jitender Singh

11 Years ago

Ans:
Hello Student,
Please find answer to your question below

$I = \int x^{x}dx$
There is not any standard mathematical expression for this integral.

What we can do is expand the integral function around zero & then do step by step integration upto few terms.
$x^{x} = x + \frac{x^{2}}{4}.(2logx)+\frac{x^{3}}{54}.(9log^{2}x-6logx+2)+........$
$I = \int (x + \frac{x^{2}}{4}.(2logx)+\frac{x^{3}}{54}.(9log^{2}x-6logx+2)+........)dx$
$I = \int xdx + \int \frac{x^{2}}{4}.(2logx)dx+\int \frac{x^{3}}{54}.(9log^{2}x-6logx+2)dx+........$ $I = \frac{x^{2}}{4} + \frac{x^{3}}{12}.(2logx)-\frac{x^{3}}{18}+\frac{x^{4}}{216}.(9log^{2}x-6logx+2)-........$ You can apply the ILATE rule furthur to get upto which term you want according to the options.