We take subst. #x=asect, so, dx=asect*tant dt#
Also, #sqrt(x^2-a^2)/x^4#
#=sqrt(a^2sec^2t-a^2)/(a^4sec^4t)#
#=(atant)/(a^4sec^4t)=1/a^3*sint/cost*cos^4t=(sintcos^3t)/a^3#
#:. I=intsqrt(x^2-a^2)/x^4dx=int(sintcos^3t)/a^3*asect*tantdt#
#=1/a^2intsintcos^3t*1/cost*sint/costdt#
#=1/a^2intsin^2tcostdt................(star)#
#=1/a^2int(sint)^2d(sint)#
#=1/a^2*(sint)^3/3#
Now, #sect=x/arArrcost=a/xrArrsint=sqrt(1-a^2/x^2)=sqrt(x^2-a^2)/x#
#:. I=1/(3a^2)*(sqrt(x^2-a^2)/x)^3=(x^2-a^2)^(3/2)/(3a^2x^3)+C#.
To proceed further from #(star)#, we can use another subst. #sint=y#,
so, #costdt=dy#, thus giving,
#I=1/a^2inty^2dy=y^3/(3a^2)=(sint)^3/(3a^2)#
