求不定积分∫√(x^2-a^2)/x^4dx（a>0)

Question

求不定积分∫√(x^2-a^2)/x^4 dx（a>0)

Answer 1

令 x=asect
√(x^2-a^2)=atant
dx=asecttantdt
∫√(x^2-a^2)/x^4 dx=∫{[a^2sect(tant)^2]/a^4(secx)^4}dt
=(1/a^2)∫(tant)^2(cost)^3dt=(1/a^2)∫[(sect)^2-1](cost)^3dt
=(1/a^2)[∫cost)dt-∫(cost)^3dt]=(1/a^2)[∫cost)dt-∫(1-(sint)^2)dsint]
=(1/3a^2)(sint)^3+c=(1/3a^2)(√(x^2-a^2)/x)^3+c