Integral of ∫ sin(4x) / cos^4(x) dx

[@MasterWuMathematics]

Links to reference videos: 1. Trig and Complex Numbers - Multiple Angle Identities: kzbin.info/www/bejne/oofEhoVpp8uLbKM

[@bopaliyaharshal2399]

Pls sir solve this Integration -∞ to ∞ ∫ ( cosx ) / ( x^(2) + 1 ) dx and don't use Feynman's trick or method then how can we solve this problem if you solve it then you are real Mathematican, can u make a video for 🙏🤔.

[@bopaliyaharshal2399]

sir I give you clues this problem 3 ways 1 way Feynman's trick 2 way inverse 3 way paradox 2 way then -∞ to ∞ ∫ f(x)dx = 1 , -∞ to ∞ ∫ ( f(x) + f (π-x) ) dx = 2 then f(x) = ?? plz solve this problem and make a video or tell me answer by replie my comment plz sir 🙏🙏 3 way we use ∞ - ∞ = π this is one paradox we use this because this paradox work all intergel questions then I apply this then If I = -∞ to ∞ ∫ ( cosx ) / ( x^(2) + 1 ) dx Then I = 1/2 (-∞ to ∞ ∫ [ π^(2)cosx - 2xπcosx ] / [ ( x^(2) + 1 ) ( (π-x)^(2) + 1 ) ] dx )

[@bopaliyaharshal2399]

-∞ to ∞ ∫ f(x)dx = 1 , -∞ to ∞ ∫ ( f(x) + f (π-x) ) dx = 2 then f(x) = ?? plz solve this problem and make a video or tell me answer by replie my comment plz sir 🙏🙏

[@MasterWuMathematics]

I think the answer is f(x) = e^(-πx^2)

[@bopaliyaharshal2399]

@@MasterWuMathematics this is part of integration Integration -∞ to ∞ ∫ ( cosx ) / ( x^(2) + 1 ) dx

[@MasterWuMathematics]

But -∞ to ∞ ∫ ( cosx ) / ( x^(2) + 1 ) dx is not equal to 1.

[@bopaliyaharshal2399]

@MasterWuMathematics I can explain I = -∞ to ∞ ∫ ( cosx ) / ( x^(2) + 1 ) dx then both side inverse (I)^(-1) = (-∞ to ∞ ∫ ( cosx ) / ( x^(2) + 1 ) dx )^(-1) (I)^(-1) = 1/ (-∞ to ∞ ∫ ( cosx ) / ( x^(2) + 1 ) dx ) (I)^(-1) =( -∞ to ∞ ∫ f(x) dx ) / (-∞ to ∞ ∫ ( cosx ) / ( x^(2) + 1 ) dx )

[@bopaliyaharshal2399]

@MasterWuMathematics Now you know why I asked that 👍