\[ I = \int_{b}^{a} x f(x) \,dx \quad \dots (1) \] \[ I = \int_{b}^{a} (a + b - x) f(a + b - x) \,dx \] \[ \left( \because \int_{b}^{a} f(x) \,dx = \int_{b}^{a} f(a + b - x) \,dx \right) \] \[ I = \int_{b}^{a} (a + b - x) f(x) \,dx \] \[ I = (a + b) \int_{b}^{a} f(x) \,dx - I \quad \text{[Using (1)]} \] \[ I + I = (a + b) \int_{b}^{a} f(x) \,dx \] \[ 2I = (a + b) \int_{b}^{a} f(x) \,dx \] \[ I = \frac{a + b}{2} \int_{b}^{a} f(x) \,dx \]