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