The Integral of sin(2x)/sin(4x)

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Evaluating the indefinite integral of sin2x/sin4x

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Here is a faster way. Make use of both the identities sin 2x = 2 sin x cos x and cos 2x = (cos x)^2 - (sin x)^2 and we have the integral // ( (2 sin x cos x)/( cos 4x)) dx and then in the
denominator make use of the identity for cos 2x and we have // ( 1 / 2 (( cos x)^2 - (sin x)^2 ) and then multiply numerator and denominator by (sec x)^2 and we finally have
(1/2) // (( sec x)^2) / (( 1 - (tan x)^2) and finally make the substitution tan x = u and (sec x)^2 dx = du and the integral becomes // ( 1 / ( 1 - u^2) ) du and the answer is
(1/2) artanh (u) + C or (1/2) artanh ( tan x) + C. Forgot to mention that the back-to-back slashes indicate the integral sign. Seems to work well. “artanh” is the inverse
hyperbolic tangent and it can be abbreviated as “artanh” or “arctan” or finally “argtanh” the last of which stands for the "inverse argument hyperbolic tangent”. No need to mess
with partial fractions for this solution, but you could proceed and evaluate //(1/(1-u*2) using partial fractions in which case your answer would be defined by logarithmic functions.
as shown in this video. Usually the are two or three or ?? way to do a particular problem all of which are correct and you can spend a lot of time attempting to show that both
answers are in fact correct. Have fun doing that if you like!!

thomasarch

Can we write sin4x as 2sin2xcos2x then we get 1/cos2x and integrating that ln|sec2x+tan2|/4?

ron

How do you find out which extra multiplications like (sec(u) + tan(u)) / (sec(u) + tan(u)) will work?

alexandre

I find the lack of subscribers to this channel disturbing. But cool video mate. I subbed yesterday when i saw your "saving monkeys with calculus" video and instantly thought: this channel is a gem. keep up the good work!

proof-xxvv

Please solve integral of sin4x ÷sin2x dx

m.s.acharsuhas

Bizarrely similar to blackpenredpen's problem published on the same day.

cycklist

Can you answer integration of tan(inverse x) × logx

harshmehta

You solve for intergal sec earlier why not apply that?

donaldey

I feel anxiety in the end sidenote.
I like the way you explain your steps, this problem is a classic for integration, but have you felt like doing other stuffs than integration (like topology, statistics, ... etc) on your YT channel?
If you prefer answering to people's question, I recommend to you the Maths discord where people often asks question like that, and it could be cool to have someone like you to help beginners.
Finally, maybe it's too personal, but what are your engineering specialities? Where did you study? What do you like in sciences?

PackSciences

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