12) [10 points] Determine which of the following definite integrals equals \( \lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{n} \sqrt{\frac{i}{n}} \).

\( \lim _ { m \rightarrow + \infty } m [ 2 - \sin ( n ^ { 2 } + 1 ) ] \)

Consider the following. \[ \begin{array}{l}y=x^{2} \\ y \\ y \\ y\end{array} \] (b) Use the integration capabilities of the graphing utility to approximate the area to four de

\( \frac{d^{4} y}{d x^{4}} \mathrm{y}=\cos 2 x \) إذا كانت

8. \( \int \frac{8 x^{2} d x}{\left(x^{3}+9\right)^{3}} \)

Find the Maclaurin series for the following function and determine its radius of convergence \( R \). \( f(x)=\ln \left(\frac{1+x}{1-x}\right) \). \( R=\square \) Use the first four terms of the series to approximate \( \ln (3) \). (Round your answer to six decimal places.) \( \ln (3) \approx \square \)