Question

Let f be a differentiable function such that f(1)=π/2 and f'(x)=3 arctan(x²-3x+2). What is the value of f(3)?

Answer 1

To find the value of f(3), we need to integrate the given derivative function and evaluate the definite integral from 1 to 3.

Step-by-step explanation:

To find the value of f(3), we need to integrate the given derivative function. Let's start by finding the antiderivative of 3 arctan(x²-3x+2). We can use the power rule and the fact that the derivative of arctan(x) is 1/(1+x²). Integrating, we have:

f(x) = 3 ∫ (arctan(x²-3x+2)) dx

Next, we evaluate the definite integral from 1 to 3:

f(3) - f(1) = 3 ∫(1 to 3) (arctan(x²-3x+2)) dx

By substituting x values and evaluating the integral, we can find the value of f(3).