Find the indefinite integral. (Note: Solve by the simplest method-not all require integration by parts. Remember the constant of integration.) \( \int x \sqrt{x-14} d x \)

To solve the integral \( \int x \sqrt{x-14} \, dx \), we can use a substitution method. Let \( u = x - 14 \), which implies \( x = u + 14 \) and \( dx = du \). Then, we rewrite the integral: \[ \int (u + 14) \sqrt{u} \, du = \int (u^{3/2} + 14\sqrt{u}) \, du \] Now we can integrate each term separately: 1. For \( u^{3/2} \), the integral is \(\frac{u^{5/2}}{5/2} = \frac{2}{5}u^{5/2}\). 2. For \( 14\sqrt{u} \), the integral is \(14 \cdot \frac{u^{3/2}}{3/2} = \frac{28}{3} u^{3/2}\). Combining these results gives us: \[ \int (u^{3/2} + 14\sqrt{u}) \, du = \frac{2}{5}u^{5/2} + \frac{28}{3}u^{3/2} + C \] Substituting back \( u = x - 14 \): \[ = \frac{2}{5}(x-14)^{5/2} + \frac{28}{3}(x-14)^{3/2} + C \] Thus, the final result is: \[ \int x \sqrt{x-14} \, dx = \frac{2}{5}(x-14)^{5/2} + \frac{28}{3}(x-14)^{3/2} + C \]