Question

求两个积分已经详细步骤是求两个积分以及详细步骤∫1/(y^2+x^2)^3/2 dx∫x/(y^2+x^2)^3/2 d

Answer 1

说明：如果y不是关于x的函数,解法如下.

设x=ytant,则sint=x/√(x²+y²),cost=y/√(x²+y²),dx=ysec²tdt

于是,有

∫1/(y²+x²)^3/2 dx=∫ysec²tdt/(y³sec³t)

=1/y²∫costdt

=sint/y²+C (C是积分常数)

=(x/√(x²+y²))/y²+C

=x/(y²√(x²+y²))+C;

∫x/(y^2+x^2)^3/2 dx=∫ytantysec²tdt/(y³sec³t)

=1/y∫sintdt

=-cost/y+C (C是积分常数)

=-(y/√(x²+y²))/y+C

=-1/(√(x²+y²))+C.