If f(x)=∫ (from -x to x) cos(t)/(1+e^t) dt and x=asin(77/85), then the absolute value of f′(x)−f(x) can be expressed as a/b, where a and b are coprime positive integers. What is the value of a+b?

To find the value of a + b, we need to calculate the absolute value of f′(x) - f(x) and express it as the fraction a/b, where a and b are coprime positive integers. Let's break down the problem step by step.

1. Start by finding f'(x), the derivative of f(x). We'll rename f(x) as F(x) to avoid confusion with the variable of integration:
F(x) = ∫[from -x to x] cos(t)/(1 + e^t) dt

To evaluate f'(x), we can use the Fundamental Theorem of Calculus:
f'(x) = d/dx [F(x)]
= cos(x)/(1 + e^x) - [cos(-x)/(1 + e^-x)]
= cos(x)/(1 + e^x) - cos(x)/(1 + e^x)
= 0

Therefore, f'(x) is zero.

2. Now, let's find f(x) using the given expression for x: x = asin(77/85). We'll substitute this value into the integral:
F(x) = ∫[from -x to x] cos(t)/(1 + e^t) dt
= ∫[from -asin(77/85) to asin(77/85)] cos(t)/(1 + e^t) dt

However, this integral cannot be evaluated directly. To proceed, we'll use a trick called the symmetry property of integrals.

3. Symmetry property: For any function f(x) that is symmetric around x = 0, we have ∫[-a to a] f(x) dx = 2∫[0 to a] f(x) dx.

In our case, the integrand cos(t)/(1 + e^t) is an even function, which means cos(t)/(1 + e^t) = cos(-t)/(1 + e^-t).

Applying the symmetry property, we can simplify the integral:
F(x) = 2∫[0 to asin(77/85)] cos(t)/(1 + e^t) dt

4. To evaluate this integral, we need to make a substitution. Let u = e^t, du = e^t dt.
When t = 0, u = e^0 = 1, and when t = asin(77/85), u = e^(asin(77/85)).

Substituting into the integral, we have:
F(x) = 2∫[1 to e^(asin(77/85))] (1/u) du
= 2 ln|u| |from 1 to e^(asin(77/85))]
= 2 ln(e^(asin(77/85))) - 2 ln(1)
= 2 asin(77/85),

where ln denotes the natural logarithm.

5. Now, we need to find the absolute value of f'(x) - f(x):
|f'(x) - f(x)| = |0 - 2 asin(77/85)|
= 2 |asin(77/85)|.

The absolute value eliminates the negative sign. Since asin(77/85) is positive, we can rewrite the expression as:
|f'(x) - f(x)| = 2 asin(77/85).

6. Finally, we have |f'(x) - f(x)| = 2 asin(77/85).
The numerator a is 2, and the denominator b is asin(77/85).

The value of a + b is 2 + 77/85 = (2 * 85 + 77)/85 = 247/85.

Therefore, the value of a + b is 247 + 85 = 332.