a) State or imply $\frac{{du}}{{dx}} = {\sec ^2}x$

Express integrand in terms of $u$ and $du$ 

Integrate to obtain $\frac{{{u^{n + 1}}}}{{n + 1}}$ or equivalent

Substitute correct limits correctly to confirm given result $\frac{1}{{n + 1}}$

b)(i) Use ${\sec ^2}x = 1 + {\tan ^2}x$ twice

Obtain integrand ${\tan ^4}x + {\tan ^2}x$

Apply result from part (a) to obtain $\frac{1}{3}$

Or

Use ${\sec ^2}x = 1 + {\tan ^2}x$ and the substitution from (a)

Obtain $\int {{u^2}du} $

Apply limits correctly and obtain $\frac{1}{3}$

(ii) Arrange, perhaps implied, integrand to

${t^9} + {t^7} + 4\left( {{t^7} + {t^5}} \right) + {t^5} + {t^3}$

Attempt application of result from part (a) at least twice

Obtain $\frac{1}{8} + \frac{4}{6} + \frac{1}{4}$ and hence $\frac{{25}}{{24}}$ or exact equivalent