Evaluating Integrals

$6$ pt$.$

$($a$)$ Determine the general antiderivative by evaluating the following indefinite integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(2x^3-6\sqrt[\phantom{2}]{x}-\frac{2}{x^2})dx$

$7$ pt$.$

$($b$)$ Determine the area under the curve across the interval $1,4$ through the evaluation of the definite integral.

${\displaystyle \underset{1}{\overset{4}{}}}(2x^3-6\sqrt[\phantom{2}]{x}-\frac{2}{x^2})dx$