This is only about, how I came up with this weird idea.
I was considering about relationship between radius(r),arc of the circle (s) and radian (θ) such that;
r/s=θ
And to convince myself completely, I wanted to try following method.
When s,ϕ are very small like δs and δϕ, we can assume that the triangle that rsr with angle δϕ that is;
So from the triangle, we can conclude that sin(δϕ2)=δs2r and while δs,δϕ go to the 0 we can take integral at both side.
Conclusion:
What does following mean?
I=∫sin(dx)
1.
I considered that what if we take the integral to inside of sin(x)?
so I=sin(x+C)
2.
I've tried definition of riemann integral.
∫baf(x)dx=limn→∞n∑k=0f(a+kb−an)(b−an)
But what is the function? f(dx) doesn't look like just f(x) or I can try following but it doesn't make any sense to me, as well.
∫f(dx)=∫f(dx)dxdx so U(x)=f(dx)dx, but I couldn't finish.
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