10. Evaluate the indefinite integral. (Use \( C \) for the constant of integration.) \[ \int e^{x} \sqrt{5+e^{x}} d x \]

To evaluate the indefinite integral \[ \int e^{x} \sqrt{5+e^{x}} \, dx, \] we can use substitution to simplify the expression. Let \[ u = 5 + e^{x}. \] Then, the differential \( du \) is \[ du = e^{x} dx \quad \Rightarrow \quad dx = \frac{du}{e^{x}} = \frac{du}{u - 5}. \] Now, rewrite the integral in terms of \( u \): \[ \int e^{x} \sqrt{5 + e^{x}} \, dx = \int \sqrt{u} \, du. \] Now, solving this integral is straightforward: \[ \int \sqrt{u} \, du = \frac{2}{3} u^{3/2} + C. \] Substituting back \( u = 5 + e^{x} \), we have: \[ \frac{2}{3} (5 + e^{x})^{3/2} + C. \] Therefore, the final result of the indefinite integral is: \[ \int e^{x} \sqrt{5+e^{x}} \, dx = \frac{2}{3} (5 + e^{x})^{3/2} + C. \]