Hello BJP Santosh,

Direct integration of root tanx is not possible. You will have to use some predefined integration forms, as well as Integration by parts formula.

take tan x=t^2

now perform substitution method

to reduce integratable function to the form:

dx=[2t/(1+t^4)]dt

Next you will have to use the predefined integration form to get

∫(1 + 1/t 2 )dt / [(t - 1/t) 2 + 2] + ∫(1 - 1/t 2 )dt / [(t + 1/t) 2 -2]

Then use integration by parts, and separating the two

The final answer will be:

(1/√2) tan -1 [(tanx - 1)/(√2tan x)] + (1/2√2) log [tanx + 1 - √(2tan x)] / [tan x + 1 + √(2tan x)] + c

Hope this helps

Refer to youtube videos in case you are not satisfied with the explanation.