Final answer:
To evaluate the integral ∫13 x f′(x)dx, we can use the second fundamental theorem of calculus to find antiderivatives and evaluate the integral using the given limits.
Step-by-step explanation:
To evaluate the integral ∫13 x f′(x)dx, we can use the second fundamental theorem of calculus, which states that if F(x) is an antiderivative of f(x), then ∫ab f(x)dx = F(b) - F(a). Since we are given the values of f''(x) and f'(x) at certain points, we can find antiderivatives of f''(x) and f'(x) to find the antiderivative of f'(x). From there, we can evaluate the integral using the given limits.
First, let's find the antiderivative, or the indefinite integral, of f''(x) which gives us f'(x). Since f''(x) is a constant, its antiderivative is a linear function. So, we have f'(x) = f''(1)(x - 1) + f'(1).
Next, we integrate f'(x) to find f(x). Since f'(x) is a linear function, its antiderivative is a quadratic function. So, we have f(x) = ⅓f''(1)(x - 1)^2 + f'(1)(x - 1) + f(1). Now, we can evaluate the definite integral ∫13 x f′(x)dx. Using the second fundamental theorem of calculus, we have ∫13 x f′(x)dx = f(3) - f(1).