$(9$ points$)$ Consider the integral $I={\displaystyle \underset{0}{\overset{1}{}}}xarctan(x)dx.$

$($a$)$ Write explicitly an approximation for I using the right$-$hand sum with $n-$slices. You can use the sigma notation or the $...$ notation. If you use notation such as $x$ or $x_k,$ be sure to explain exactly what it means.

$($b$)$ Find a number of slices for which your approximation in part $($a$)$ is smaller than ${10}^{-6}.$ Hint: the error bound formula for the right$-$hand sum is where $M_{(1)}$ is an upper bound for $|f^{\text{'}}(x)|$ on $a,b.$

$($c$)$ Find a power series representation for ${\displaystyle \underset{0}{\overset{1}{}}}xarctan(x)dx.$ You can use the sigma notation or the $...$ notation. Hint: the derivative of $arctan(x)$ is $\frac{1}{1+x^2}.$

$($d$)$ Use the result in part $($c$),$ how many $($nonzero$)$ terms does one need in order to estimate ${\displaystyle \underset{0}{\overset{1}{}}}xarctan(x)dx$ with an error of magnitude less than ${10}^{-6}?$ Explain your reasoning.

$($e$)$ Which method, $($a$)$ the right$-$hand sum or $($c$)$ the power series representation, do you prefer? There is no correct answer. We just want to know which method you'd like to use!