Question

Integrate the following w.r.t.x :  `sec^2x sqrt(7 + 2 tan x - tan^2 x)`

Answer 1

Let I = `int sec^2x sqrt(7 + 2 tan x - tan^2x) *dx`
Put tan x = t
∴ sec²x·dx = dt

∴ I = `int sqrt(7 + 2t - t^2)*dt`

= `int sqrt(7 - (t^2 - 2t))*dt`

= `int sqrt(8 - (t^2 - 2t + 1))*dt`

= `int sqrt((2sqrt(2))^2 - (t - 1)^2)*dt`

= `((t - 1)/2) sqrt((2sqrt(2))^2 - (t - 1)^2) + ((2sqrt(2))^2)/(2) sin^-1((t - 1)/(2sqrt(2))) + c`

= `((t - 1)/2) sqrt(7 + 2t - t^2) + 4sin^-1 ((t - 1)/(2sqrt(2))) + c`

= `((tanx - 1)/2)sqrt(7 + 2tanx - tan^2x) + 4sin^-1 ((tanx - 1)/(2sqrt(2))) + c`.