Partial Fractions

STEP 2003, Math I, #6: Evaluate the following integrals, in the different cases that arise according to the

value of the positive constant

.

(i)

(ii)

in terms of exponentials.

By using partial fractions, prove that

(ii) Find, expressing your answers in terms of hyperbolic functions,

and

STEP 2004, Math III, #1: Show that

and find

Hence show that

By substituting

in this result, or otherwise, find