Question 10 - A-Level Maths

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2011
Session	November
Topic	Integration by Substitution

Use the substitution $u = \tan x$ to show that, for $n \neq - 1$, $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { n + 2 } x + \tan ^ { n } x \right) \mathrm { d } x = \frac { 1 } { n + 1 }$$
Hence find the exact value of
(a) $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sec ^ { 4 } x - \sec ^ { 2 } x \right) \mathrm { d } x$,
(b) $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 9 } x + 5 \tan ^ { 7 } x + 5 \tan ^ { 5 } x + \tan ^ { 3 } x \right) \mathrm { d } x$.