$\displaystyle \lim _ {n \rightarrow \infty} {} \int _ {0} ^ {\frac{1} {2}} {} \left (x+x ^ {3} + \cdots +x ^ {2n+1} \right) dx$ 의 값은?

• ①$\ln 2 \sqrt{3}$
• ②$\displaystyle \ln \frac{5 \sqrt{3}} {3}$
• ③$\displaystyle \ln \frac{4 \sqrt{3}} {3}$
• ④$\ln \sqrt{3}$
• ⑤$\displaystyle \ln \frac{2 \sqrt{3}} {3}$