a) Either

Use integration by parts and reach an expression $k{x^2}{\text{ }}lnx{\text{ }} \pm n\int {{x^2}.\frac{1}{x}\,dx} $

Obtain $\frac{1}{2}{x^2}ln{\text{ }}x - \int {\frac{1}{2}x\,dx} $ or equivalent

Obtain $\frac{1}{2}{x^2}ln{\text{ }}x - \frac{1}{4}{x^2}$

Or

Use Integration by parts and reach an expression $kx\left( {x\,ln\,x - x} \right) \pm m\int {xln\,x - xdx} $

Obtain $I = \left( {{x^2}lnx - {x^2}} \right) - I + \int {xdx} $

Obtain $\frac{1}{2}{x^2}ln{\text{ }}x - \frac{1}{4}{x^2}$

Substitute limits correctly and equate to 22, having integrated twice

Rearrange and confirm given equation $\alpha  = \sqrt {\frac{{87}}{{2ln{\text{ }}\alpha  - 1}}} $

b) Use iterative process correctly at least once

Obtain final answer $5.86$

Show sufficient iterations to 4 d.p. to justify 5.86 or show a sign change in the interval (5.855, 5.865)

$\left( {6 \to 5.8030 \to 5.8795 \to 5.8491 \to 5.8611 \to 5.8564} \right)$