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Question

1+tan x tan x+θ dx

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Solution

Let I=1+tan x tan x+θdx = 1+tanxtan x+tan θ1-tan x tan θdx = 1+tan2 x1-tan x tan θdx = sec2x dx1-tan x tan θPutting tanx=t sec2x=dtdx dx=dtsec2x I= 11-t tanθdt = -1tan θ ln 1-t tan θ+C 1ax+bdx=1aln ax+b+C = -cot θ ln 1-tanx tan θ+C = cot θ ln 11-tan x tan θ+C = cot θ ln cosx cosθcos x cos θ-sin x sin θ+C = cot θ ln cos xcos x+θ+C' Let C'=C+cot θ ln cosθ

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