a$)$ Consider the function $arctan(x^2).$

Write a partial sum for the power series which represents this function consisting of the first $4$ nonzero terms. For example, if the series were ${\displaystyle \underset{n=0}{\overset{}{}}}{3}^n{x}^{2n},$ you would write $13x^2{3}^2{x}^4{3}^3{x}^6.$ Also indicate the radius of convergence.

Partial Sum: $$

Radius of Convergence:

b$)$ Use part a$)$ to write the partial sum for the power series which represents ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}arctan(x^2)dx.$

Write the first $4$ nonzero terms. Also indicate the radius of convergence.

Partial Sum: $$

Radius of Convergence:

c$)$ Use part b$)$ to approximate the integral ${\displaystyle \underset{0}{\overset{0\mathrm{.}2}{}}}arctan(x^2)dx.$