Substitution. Consider the indefinite integral:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^3}{\sqrt[\phantom{2}]{x^2+1}}dx$

Find $(using$ one $of$ the substitutions below$)an$ antiderivative $as$ a function $ofx.$ Also, perform each

substitution $to$ produce $an$ equivalent integral. Show the details $of$ each substitution including the

differential, and give the simplified transformed integral $(not$ evaluated.$)$

ANTIDERIVATIVE:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^3}{\sqrt[\phantom{2}]{x^2+1}}dx=$

WORK $(Finding$ the antiderivative from one $of$ the substitutions$)$:

$X1.(Continued)$

SUBSTITUTION $1$: $u=x^2+1$

TRANSFORMED INTEGRAL:

DETAILS:

SUBSTITUTION $2$: $v=\sqrt[\phantom{2}]{x^2+1}$

TRANSFORMED INTEGRAL:

DETAILS:

SUBSTITUTION $3$: $tan()=x$

TRANSFORMED INTEGRAL:

DETAILS: