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Short answer question
1. (a) Starting with the equation
x = tanh y
show that
y = tanh-1 x = 1/2 loge(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 eiθ = cos θ + i sinθ to show that
sinh(iθ) = i sin(θ)
(d) Use the definitions
cosh(x) = 1/2(ex + e-x, sinh(x) = 1/2(ex - 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 In-2(x).
(b) Hence show that
Π/4∫3Π/4 1/sin4(x) dx = 8/3
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