Contour integration

P_n(x)=(1/pi)int{0}{pi}{(x+i*sqrt{1-x^2}cos theta)^n}d theta

where -1≤x≤1

since

delim{|}{x+i*sqrt{1-x^2}cos theta }{|}=sqrt{x^2+(1-x^2)cos^2 theta}

sqrt{x^2+(1-x^2)cos^2 theta}sqrt{x^2+(1-x^2)}=1

it follows that

delim{|}{ P_n(x)}{|}(1/pi)int{0}{pi}{delim{|}{x+i*sqrt{1-x^2}cos theta}{|}^n}d theta

(1/pi)int{0}{pi}d theta = 1