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Real variables x and y...

Real variables x and y are related by the equation

3 ln(y + 4) = 2 ln(x + 2) - 2 ln(x + 9) + 3 ln(x^2 - 1).

(i) Determine the range of values of x and y for which the expressions on

each side of this equation are de?ned.

(ii)

Find y explicitly as a function of x, that is, express the equation in the

form y = f(x), simplifying your answer as far as possible.

Show that the expressions

v3 cos(5t + p/6) and (v3/2)(v3 cos(5t) - sin(5t)) are equivalent.

(iii)

If x tan y - y tan x = 1, use implicit di?erentiation to determine dy/dx,

expressing your answer in the form

dy/dx= f(x, y),

that is, an expression that involves terms in both x and y.

(iv)

determine the inde?nite integral of the function

f(x) = 1/ (x - 1)(x - 2)

where (1 < x < 2).

(v)

Evaluate

integral sign (top)p/2

(bottom)3p/8

sec(2x) dx

3 ln(y + 4) = 2 ln(x + 2) - 2 ln(x + 9) + 3 ln(x^2 - 1).

(i) Determine the range of values of x and y for which the expressions on

each side of this equation are de?ned.

(ii)

Find y explicitly as a function of x, that is, express the equation in the

form y = f(x), simplifying your answer as far as possible.

Show that the expressions

v3 cos(5t + p/6) and (v3/2)(v3 cos(5t) - sin(5t)) are equivalent.

(iii)

If x tan y - y tan x = 1, use implicit di?erentiation to determine dy/dx,

expressing your answer in the form

dy/dx= f(x, y),

that is, an expression that involves terms in both x and y.

(iv)

determine the inde?nite integral of the function

f(x) = 1/ (x - 1)(x - 2)

where (1 < x < 2).

(v)

Evaluate

integral sign (top)p/2

(bottom)3p/8

sec(2x) dx

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