Let f(x) = 12 * (e^(3x) - 3*e^x) / (e^(2x) - 1) be defined for x 0, and let g(x) be the inverse of f(x). If the integral from 8 to 27 of g(x) dx = a*ln(3) - b*ln(2) - c, find the value of a - (b + c).

Question

Let f(x) = 12 * (e^(3x) - 3*e^x) / (e^(2x) - 1) be defined for x > 0, and let g(x) be the inverse of f(x). If the integral from 8 to 27 of g(x) dx = a*ln(3) - b*ln(2) - c, find the value of a - (b + c).

Accepted Answer

7. Simplify f(x): divide numerator and denominator by e^x: f(x) = 12*(e^(2x) - 3)/(e^(2x) - 1). Let u = e^(2x): f = 12*(u-3)/(u-1) = 12*[1 - 2/(u-1)]. Since g is the inverse of f, use the formula integralₐ^b g(x)dx = b*g(b) - a*g(a) - integral_(g(a))^(g(b)) f(t)dt.

Let f(x) = 12 * (e^(3x) - 3e^x) / (e^(2x) - 1) be defined for x > 0, and let g(x) be the inverse of f(x). If the integral from 8 to 27 of g(x) dx = aln(3) - b*ln(2) - c, find the value of a - (b + c).

Solution

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