\( I = \int _ { 0 } ^ { 1 } x f ^ { \prime } ( x ) d x \)

To evaluate the integral \[ I = \int_{0}^{1} x f^{\prime}(x) \, dx, \] we can use **integration by parts**. Integration by parts is based on the formula: \[ \int u \, dv = uv - \int v \, du, \] where \( u \) and \( dv \) are continuously differentiable functions on the interval of integration. ### Step 1: Choose \( u \) and \( dv \) Let's assign: \[ \begin{cases} u = x & \Rightarrow \frac{du}{dx} = 1 \, \Rightarrow \, du = dx, \\ dv = f^{\prime}(x) \, dx & \Rightarrow v = f(x). \end{cases} \] ### Step 2: Apply Integration by Parts Substituting into the integration by parts formula: \[ I = \int_{0}^{1} x f^{\prime}(x) \, dx = \left. x f(x) \right|_{0}^{1} - \int_{0}^{1} f(x) \cdot 1 \, dx. \] ### Step 3: Evaluate the Boundary Terms Compute the boundary term \( \left. x f(x) \right|_{0}^{1} \): \[ \left. x f(x) \right|_{0}^{1} = (1 \cdot f(1)) - (0 \cdot f(0)) = f(1) - 0 = f(1). \] ### Step 4: Combine the Results Putting it all together: \[ I = f(1) - \int_{0}^{1} f(x) \, dx. \] ### Final Result Thus, the integral simplifies to: \[ \int_{0}^{1} x f^{\prime}(x) \, dx = f(1) - \int_{0}^{1} f(x) \, dx. \] This result expresses the original integral in terms of the function \( f \) evaluated at the upper limit and the integral of \( f \) itself over the interval \([0, 1]\).