Proving the relationship between the effective value and the peak value of the sinusoidal alternating current by integration Such as the title

Proving the relationship between the effective value and the peak value of the sinusoidal alternating current by integration Such as the title

Let I = asin (2 π T / T)
According to the valid value definition, there are
In one cycle
T
RTI ^ 2 = ∫ R [asin (2 π T / T)] ^ 2DT [0, t] integral
0
T
Ti ^ 2 = (a ^ 2) ∫ [sin (2 π T / T)] ^ 2DT [0, t] integral
0
Substitution x = (2 π T / T)
π/2
Integral in I ^ 2 = 4 (a ^ 2) (1 / 2 π) ∫ [sin (x)] ^ 2DX [0, π / 2]
0
According to the integral formula
π/2
In=∫(sinx)^ndx=[(n-1)/n]I_ (n-1)
0
π/2
There is ∫ [sin (x)] ^ 2DX = (1 / 2) (π / 2) = π / 4
0
I^2 = (A^2)/2
I=A(√2)/2
It's over