Given : 2x2+2x+n>9+sin−1(sin(−1))+cos−1(cos(−1))
We know,
sin−1(sinx)=x,−π2≤x≤π2,
cos−1(cosx)=−x,−π≤x≤0
So inequality becomes :
2x2+2x+n>9−1−(−1)
⇒2x2+2x+n−9>0 is true ∀x∈R
Hence, D<0 as coefficient of x2>0
⇒4−8(n−9)<0
⇒n>192
So, minimum integral value of n=10