1/(a^2+x^2)^3/2 不定积分

此题要分两种情况
(1)当a=0时,
原式=∫dx/x³
=-1/(2x²)+C (C是积分常数)
(2)当a≠0时,
设x=atant
则dx=asec²tdt
sint=xcost/a
=(x/a)(1/sect)
=(x/a)[1/√(1+tan²t)]
=(x/a)[a/√(a²+x²)]
=x/√(a²+x²)
∴原式=∫asec²tdt/(a³sec³t)
=1/a²∫dt/sect
=1/a²∫costdt
=sint/a²+C (C是积分常数)
=x/[a²√(a²+x²)]+C (C是积分常数).