Question

Find: `int sec^2 x /sqrt(tan^2 x+4) dx.`

Answer 1

`I = int sec^2 x /sqrt(tan^2 x+4) dx.`

`"Let" tan x = t`

`sec^2 xdx = dt`

So, `I = int dt/sqrt(t^2 + 4)`

or, `I = int dt/sqrt(t^2 + 2^2)`

Since, we Know

`int dx/sqrt(x^2 + a^2) = "In" |x + sqrt(a^2 + x^2)| + "C"`

`I = "In" |t + sqrt(t^2 + 4)| + "C"`

`i.e`

`I = "In"|tanx + sqrt(tan^2 x + 4)| + "C"`