Question 4 - A-Level Maths

Starting from the definitions of $\sinh x$ and $\cosh x$ in terms of exponentials, prove that $$1 + 2 \sinh ^ { 2 } x = \cosh 2 x$$
Solve the equation $$2 \cosh 2 x + \sinh x = 5 ,$$ giving the answers in an exact logarithmic form.
Show that $\int _ { 0 } ^ { \ln 3 } \sinh ^ { 2 } x \mathrm {~d} x = \frac { 10 } { 9 } - \frac { 1 } { 2 } \ln 3$.
Find the exact value of $\int _ { 3 } ^ { 5 } \sqrt { x ^ { 2 } - 9 } \mathrm {~d} x$.