Compute the derivative to obtain an equation for cosh3x in

Question

Short answer question

1. (a) Starting with the equation

x = tanh y

show that

y = tanh^-1 x = 1/2 log_e(1+x)/(1-x)

(b) Compute the following derivative

d/dx(sinh^-1(cos(s)))

(c) Assume θ is a real number. Then use Euler's formula e^iθ = cos θ + i sinθ to show that

sinh(iθ) = i sin(θ)

(d) Use the definitions

cosh(x) = 1/2(e^x + e^-x, sinh(x) = 1/2(e^x - e^-x)

to obtain an equation for cosh(3x) in terms of cosh(x) and sinh(x).

Detailed answer question

2. (a) Use integration by parts to express

In(x) = sinⁿ(x) dx

in terms of I_n-2(x).

(b) Hence show that

_Π/4∫^3Π/4 1/sin⁴(x) dx = 8/3