SHM energies

sadhu

in SHM
average K.E when done w.r.t to time is equal to average potential energy calculated w.r.t to time.

but today in class when my sir asked me to to prove average K.e = average P.E
I just tried integrating it w.r.t to displacement and then divided it with A ,
I thought this gives us av . energies in 1/4 vibration and as all the four parts are identical
average will remain same , but i found that av . K.E=2*av. P.E

I then done this again w.r.t to time and got the answer

but still I am confused , about how can that come i.e K=2*P.E