When first learning integration techniques, people tend to find a favorite and stick with it; however, you would be surprised at how many methods will work on a given problem. Consider the following integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^3}{\sqrt[\phantom{2}]{x^2+9}}dx$

a$.$ Trigonometric Substitution: Rewrite the integral in terms of $$ by using a trigonometric substitution $($typing "theta" without the quotes will give you the variable $).$ Simplify your integrand completely, but leave it in terms of $.$ Do not evaluate the integral.

$$

$d$

b$.$ Radical Substitution: Rewrite the integral in terms of $u$ by using a radical substitution. Simplify the integrand completely, but leave it in terms of $u.$ Do not evaluate the integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}du$

c$.$ Integration By Parts: Rewrite the integral as the result of Integration By Parts by letting $dv=\frac{x}{\sqrt[\phantom{2}]{x^2+9}}dx.$ Your answer will include two parts: a term without an integral and a term with an integral $($simplify the integrand completely$).$ Do not evaluate the integral.

$-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\sqrt[\phantom{2}]{dx}$

d$.$ Standard $u$ Substitution: Rewrite the integral in terms of $u$ by using a standard, non$-$radical $u$ substitution. Simplify your integrand completely. Do not evaluate the integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\sqrt[\phantom{2}]{du}$

e$.$ Finally, evaluate the integral by using any of the methods above.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^3}{\sqrt[\phantom{2}]{x^2+9}}dx=$