The thread you cited contains the entire answer. So it's hard to guess, which part of it you didn't understand. That's why we have a homework forum and insist on using the template, because otherwise, people have to guess what you might have meant.askor said:This is the continuation of the below thread:
https://www.physicsforums.com/threads/what-is-integral-tan-2x-dx.856530/
Can someone please tell me how to integrate tan 2x dx?
The integration formula for tan 2x is ∫tan 2x dx = 1/2 ln |cos 2x| + C
To integrate tan 2x using substitution, let u = 2x, then du = 2dx. Substitute these values into the integration formula to get ∫tan u du = 1/2 ln |cos u| + C. Finally, replace u with 2x to get the final answer.
Yes, you can also use the trigonometric identity tan 2x = 2tan x / (1-tan^2 x). Rewrite the integral as ∫2tan x / (1-tan^2 x) dx and use the substitution u = tan x to get ∫2u / (1-u^2) du = ln |1-u^2| + C. Replace u with tan x to get the final answer.
Yes, you can use integration by parts to integrate tan 2x, but it may require multiple iterations. Start by letting u = tan 2x and dv = dx. Then, du = 2sec^2 2x dx and v = x. Use the integration by parts formula ∫u dv = uv - ∫v du to get ∫tan 2x dx = xtan 2x - ∫ 2x sec^2 2x dx. You can then apply integration by parts again to the remaining integral to get the final answer.
Yes, if the limits of integration are symmetrical about the origin (i.e. both positive and negative values), you can simplify the integration by using the property tan(-x) = -tan x. This will eliminate the constant of integration and give you a simpler answer.