Question

积分题求解答

第二部到第三步的具体运算过程

Answer 1

let
x=Rtanu
dx=R(secu)^2 du
x=0, u=0
x=L/2 , u=arctan[L/(2R)]
∫(0->L/2) dx/(x^2+R^2)^(3/2)
=∫(0->arctan[L/(2R)]) R(secu)^2 du/[R^3.(secu)^3]
=(1/R^2)∫(0->arctan[L/(2R)]) cosu dx
=(1/R^2) [sinu]|(0->arctan[L/(2R)])
=(1/R^2) [ L/√(4R^2+L^2)]
=L/[2R^2.√(R^2+(L/2)^2)]

Answer 2

如图，积分过程如图所示的