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Question
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Solution
Let I = `int sec^2x.sqrt(tan^2x + tan x - 7)`
Put tan x = t
∴ sec2x.dx = dt
∴ I = `int sqrt(t^2 + t - 7).dt`
= `int sqrt(t^2 + t + 1/4 - 29/4).dt`
= `int sqrt((t + 1/2)^2 - (sqrt(29)/2)^2).dt`
= `((t + 1/2)/2) sqrt((t + 1/2)^2 - 29/4) - ((29/4))/(2)log|(t + 1/2) + sqrt((t + 1/2)^2 - 29/4)| + c`
= `((2t + 1))/(4)sqrt(t^2 + t - 7) - (29)/(8)log|(t + 1/2) + sqrt(t^2 + t - 7)| + c`
= `((2tanx + 1)/4)sqrt(tan^2x + tanx - 7) - (29)/(8)log|(tanx + 1/2) + sqrt(tan^2x + tanx - 7)| + c`.
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