b): ∫(sec^2x/tanx(tan^2x-1)^(1/2) )dx
let tanx =t
differentiating both side
we get
(sec^2x) dx = dt
by puting these values in above integral
we get
=∫dt/(t(t^2 – 1)^(1/2))
we know that
d( – cosec^(-1)x)/dx = 1/x((x^2) -1)^(1/2)........................(1)
or
d(sec^(-1)x)/dx = 1/x((x^2) -1)^(1/2)..................................(2)
by using one of both
we get
{ - cosec^(-1)t +c.........................................t = tanx ( when we use ….1....... ) }
so,,
{ – cosec^(-1)tanx + c }
or when we use...........................2......
sec^(-1)t + c t = tanx
{ sec^(-1)tanx +c }
its ur required solution
