Consider the following function.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^4}{\sqrt[\phantom{2}]{1+x^2}}dx$

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

Use $x=sin(),$ where $-\frac{}{2}\frac{}{2},$ since the integrand contains the expression $\sqrt[\phantom{2}]{1+x^2}.$

Use $x=tan(),$ where since the integrand contains the expression $\sqrt[\phantom{2}]{1+x^2}.$

Use $x=sec(),$ where $0<mfrac><mrow><mi></mi></mrow><mrow><mn>2</mn></mrow></mfrac>$ or since the integrand contains the expression $\sqrt[\phantom{2}]{1+x^2}.$

$($b$)$ Apply the substitution to transform the integral into a trigonometric integral. Do rifte evaluate the integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^4}{\sqrt[\phantom{2}]{1+x^2}}dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(,)d$