Question 4 - A-Level Maths

Exam Board	Edexcel
Module	FP3 (Further Pure Mathematics 3)
Year	2009
Session	June
Topic	Integration using inverse trig and hyperbolic functions

Given that $y = \operatorname { arsinh } ( \sqrt { } x ) , x > 0$,
1. find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving your answer as a simplified fraction.
2. Hence, or otherwise, find
$$\int _ { \frac { 1 } { 4 } } ^ { 4 } \frac { 1 } { \sqrt { [ x ( x + 1 ) ] } } \mathrm { d } x$$ giving your answer in the form $\ln \left( \frac { a + b \sqrt { } 5 } { 2 } \right)$, where $a$ and $b$ are integers.