Final answer:
To solve the integral using the substitution method, we can substitute u = x² and then apply the corresponding substitutions for limits and variables. After simplifying the expression, we can integrate it to find the solution.
Step-by-step explanation:
To solve the integral ∫[0 to 3] √(x)(9 - x²) x using the substitution method with u-values ranging from 0 to 3, we can let u = x². Then, du/dx = 2x, which implies dx = du/(2x). Substituting the values into the integral, we have ∫[0 to 3] √(x)(9 - x²)x dx = ∫[0 to 3] √(u)(9 - u) du/(2x).
Now, we can proceed with the substitution by replacing the integral limits and dx with the corresponding u-values and du expression: ∫[0 to 3] √(u)(9 - u) du/(2√(u)).
Simplifying further, we get (1/2)∫[0 to 3] √(u)(9 - u) du/√(u). Finally, integrating, we have (1/2)∫[0 to 3] (u^(1/2))(9 - u) du.