Consider the following function.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^3}{\sqrt[\phantom{2}]{25-x^2}}dx$

$(a)$ Determine $an$ appropriate trigonometric substitution.

Use $x=5sin(),$ where $-\frac{}{2}\frac{}{2},$ since the integrand contains the expression $\sqrt[\phantom{2}]{5^2-x^2}.$

Use $x=5tan(),$ where since the integrand contains the expression $\sqrt[\phantom{2}]{5^2-x^2}.$

Use $x=5sec(),$ where since the integrand contains the expression $\sqrt[\phantom{2}]{5^2-x^2}.$

$(b)$ Apply the substitution $to$ transform the integral into a trigonometric integral. $Do$ not evaluate the integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^3}{\sqrt[\phantom{2}]{25-x^2}}dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(,)d$