Use fundamental theorem of calculus
Since 0≤x2 and 0≤x4 for all x∈R,
Adding 1 to both sides of the above inequality,
Also, 0≤x4≤x4+x2+1 for all x∈R.
Hence, divide x4+x2+1 to the above inequality
0≤x4x4+x2+1≤1 for all x∈R.
Since x2>0, for all x∈5, 10, divide x2 to the above inequality
0≤x2x4+x2+1≤1x2 for all x∈5, 10.
Integrating between limits 5 to 10
∫5100dx≤∫510x2x4+x2+1 dx≤∫5101x2 dx Ant derivate of 1x2 is -1x. Thus using fundamental theorem of calculus,
That is 0≤∫510x2x4+x2+1 dx≤-110--15