급수와 정적분

$\displaystyle \lim\limits _{n \rightarrow \infty} {\frac{1}{n ^{2}} \sum\limits _{k = 1} ^{n} kf \left( \frac{k}{2 n} \right) =}$$\displaystyle \lim\limits _{n \rightarrow \infty} {\frac{1}{n} \sum\limits _{k = 1} ^{n} \frac{k}{n} f \left( \frac{k}{2 n} \right)}$ $\displaystyle = \int _{0} ^{1} x f \left( \frac{x}{2} \right) dx$$\displaystyle = \int _{0} ^{1} x \left( x ^{2} + \frac{a}{2} x \right) dx$ $\displaystyle = \int _{0} ^{1} \left( x ^{3} + \frac{a}{2} x ^{2} \right) dx$$\displaystyle = \left[ \frac{x ^{4}}{4} + \frac{a}{6} x ^{3} \right] _{0} ^{1}$ $\displaystyle = \frac{1}{4} + \frac{a}{6} = 2$, $\therefore$ $\displaystyle a = \frac{21}{2}$