求积分符下dx/(4+x^2)^(3/2) 如题

来源:学生作业帮助网 编辑:作业帮 时间:2024/05/12 12:41:09
求积分符下dx/(4+x^2)^(3/2) 如题

求积分符下dx/(4+x^2)^(3/2) 如题
求积分符下dx/(4+x^2)^(3/2)
如题

求积分符下dx/(4+x^2)^(3/2) 如题
令x=2/u,则:u=2/x,dx=-(2/u^2)du.
∴(4+x^2)^(3/2)=(4+4/u^2)^(3/2)=8(1+u^2)√(1+u^2)/u^3,
∴∫[(4+x^2)^(3/2)]dx
=-(1/8)∫{1/[(1+u^2)√(1+u^2)/u^3]}(2/u^2)du
=-(1/4)∫{u/[(1+u^2)√(1+u^2)]}du
=-(1/8)∫{1/[(1+u^2)√(1+u^2)]}d(1+u^2)
=-(1/8)∫[(1+u^2)^(-3/2)]d(1+u^2)
=-(1/8)×[1/(-3/2+1)](1+u^2)^(-3/2+1)+C
=(1/4)(1+u^2)^(-1/2)+C
=1/[4√(1+u^2)]+C
=1/{4√[1+(2/x)^2]}+C
=x/[4√(x^2+4)]+C.

(4+x^2)^(5/2)/5+c