Question

integral of 1+tanxtan[x+a]

Answer 1

Hence it is the answer

Answer 2

Answer:

cot θ ln cos xcos x+θ+C+cot θ ln cosθ

Step-by-step explanation:

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 θ

Putting 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