求不定积分∫dx/（x^2+a^2)^3/2

Question

求不定积分∫dx/（x^2+a^2)^3/2

Answer 1

∫dx/（x²+a²)^3/2,x=atant,dx=asec²tdt
=a∫[sec²t/(a²tan²t+a²)^3/2]dt
=a∫[sec²t/a³sec³t]dt
=(1/a²)∫cosdt
=(1/a²)*sint+C
=(1/a²)*[x/√(x²+a²)]+C,C为任意常数