求(x*dx)/[(a^2+x^2)^(3/2)]0到正无穷和0到R的积分结果（a为常数）最好有过程

Question

求(x*dx)/[(a^2+x^2)^(3/2)]0到正无穷和0到R的积分结果（a为常数）最好有过程

Answer 1

令 x=a tant,(a^2+x^2)^(3/2)= a^3 (sect)^3,dx= a (sect)^2dt
1.∫(x*dx)/[(a^2+x^2)^(3/2)]
= (1/a)∫sint dt t从0到π/2
= 1/a
2.I = (1/a)∫sint dt t从0到arctan(R/a)
= (-1/a) [ 1- cos(arctan(R/a))]
= 1/a – 1/√(a^2+R^2)