Question

Let f be a differentiable function such that f(1)=pi/2 and f'(x)=3 arctan(x^2-3x+2). What is the value of f(3)?

Answer 1

Since f is differentiable, we can use the fundamental theorem of calculus to calculate the value of f(3). The fundamental theorem of calculus states that if f is a continuous function with a continuous derivative on an interval [a, b], then for any x in [a, b],

f(x) = f(a) + ∫_a^x f'(t) dt

Using this formula, we have:

f(3) = f(1) + ∫_1^3 f'(t) dt
= pi/2 + ∫_1^3 3 arctan(t^2 - 3t + 2) dt

The integral on the right-hand side can be calculated using substitution, but the exact result can be quite complicated. An approximate numerical value for f(3) can be obtained using a suitable numerical integration method.