IIT-JEE 2008 · Paper 1, Mathematics question 9

Riemann sums. Write $\dfrac n{n^2+kn+k^2}=\dfrac1n\cdot\dfrac1{1+\frac kn+\left(\frac kn\right)^2}$. With $f(x)=\dfrac1{1+x+x^2}$:
$$\int_0^1f(x)\,dx=\frac2{\sqrt3}\left[\tan^{-1}\frac{2x+1}{\sqrt3}\right]_0^1=\frac2{\sqrt3}\left(\frac\pi3-\frac\pi6\right)=\frac\pi{3\sqrt3}.$$

$f$ is decreasing on $[0,1]$, so the right-endpoint sum $S_n$ ($k=1,\ldots,n$) underestimates the integral and the left-endpoint sum $T_n$ ($k=0,\ldots,n-1$) overestimates it.

(A), (D)