Question 5 - A-Level Maths

Exam Board	OCR
Module	FP2 (Further Pure Mathematics 2)
Year	2006
Session	June
Topic	Integration using inverse trig and hyperbolic functions

Express $t ^ { 2 } + t + 1$ in the form $( t + a ) ^ { 2 } + b$.
By using the substitution $\tan \frac { 1 } { 2 } x = t$, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 + \sin x } \mathrm {~d} x = \frac { \sqrt { 3 } } { 9 } \pi$$