Questions: Evaluate ∫[f(x) · g(x)]′ dx

Solution Steps

To evaluate the integral of the derivative of the product of two functions, we can use the fundamental theorem of calculus. The integral of the derivative of a function is simply the function itself plus a constant of integration.

Let \( f(x) \) and \( g(x) \) be two differentiable functions. We are interested in evaluating the integral of the derivative of their product, \( \int [f(x) \cdot g(x)]' \, dx \).

Using the product rule of differentiation, we have: \[ [f(x) \cdot g(x)]' = f(x) \cdot g'(x) + g(x) \cdot f'(x) \]

Now, we can integrate the expression obtained from the product rule: \[ \int [f(x) \cdot g(x)]' \, dx = f(x) \cdot g(x) + C \] where \( C \) is the constant of integration.

Final Answer