Question 4 - A-Level Maths

Prove, from definitions involving exponentials, that $$\cosh 2 u = 2 \cosh ^ { 2 } u - 1$$
Prove that $\operatorname { arsinh } y = \ln \left( y + \sqrt { y ^ { 2 } + 1 } \right)$.
Use the substitution $x = 2 \sinh u$ to show that $$\int \sqrt { x ^ { 2 } + 4 } \mathrm {~d} x = 2 \operatorname { arsinh } \frac { 1 } { 2 } x + \frac { 1 } { 2 } x \sqrt { x ^ { 2 } + 4 } + c$$ where $c$ is an arbitrary constant.
By first expressing $t ^ { 2 } + 2 t + 5$ in completed square form, show that $$\int _ { - 1 } ^ { 1 } \sqrt { t ^ { 2 } + 2 t + 5 } \mathrm {~d} t = 2 ( \ln ( 1 + \sqrt { 2 } ) + \sqrt { 2 } )$$ \section*{[Question 5 is printed overleaf.]}