Integrating hyperbolic and inverse functions

Solve the differential equation

given that y = 3 when x = ln2

Give your answer in an exact form.

By making a suitable substitution, show that

The hyperbola H has equation y² x² 16

The circle C has equation x² y² 32

The diagram below shows part of the graph of H and part of the graph of C.

Show that the shaded region in the first quadrant enclosed by H, C, the x-axis and the y-axis has area

The integral S_n is defined by

Show that for n 2

Hence show that

Use the definition of the cosh function to prove that

Ronald says that this contradicts the result given in part (a).

Explain why Ronald is wrong.

A curve has equation y cosh x

Show that the arc length of the curve from x a to x b, where 0 a b, is equal to