sarthak goel 25th Nov, 2019

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.