You are asking us to do this for you. NO WAY!
Note that $\displaystyle f(x)=\frac{x^2}{x^6+1}$ is an even function.
So the question reduces to $\displaystyle 2\int_0^1 {f(x)dx} $.
To partition the interval $\displaystyle [0,1]$ into $\displaystyle n$ regular subdivision we get $\displaystyle \Delta_x=\frac{1}{n}$.
Then $\displaystyle x_k=k\cdot\Delta_x,~k=0,~1,\cdots,~n$.
The midpoints are $\displaystyle m_k=\frac{x_k-x_{k-1}}{2},~k=1.\cdots,n$.
Now the midpoint sum is $\displaystyle \sum\limits_{k = 1}^n {f\left( {m_k } \right)\Delta _x } $