Step-by-step Solution:

To solve the integral \( I = \int \sqrt{x} e^{\sqrt{x}} dx \), we can proceed with the following steps: 1. Substitution: Let \( \sqrt{x} = t \). Then, \( x = t^2 \) and \( dx = 2t \, dt \). 2. Rewrite the Integral: Substitute \( \sqrt{x} = t \) and \( dx = 2t \, dt \) into the integral: \[ I = \int t e^{t} \cdot 2t \, dt = \int 2t^2 e^{t} \, dt \] 3. Integration by Parts: Use integration by parts, where \( u = 2t^2 \) and \( dv = e^{t} \, dt \). Then, \( du = 4t \, dt \) and \( v = e^{t} \). Apply the integration by parts formula: \[ I = 2t^2 e^{t} - \int 4t e^{t} \, dt \] 4. Integrate \( \int 4t e^{t} \, dt \): Use integration by parts again, where \( u = 4t \) and \( dv = e^{t} \, dt \). Then, \( du = 4 \, dt \) and \( v = e^{t} \). Apply the integration by parts formula: \[ \int 4t e^{t} \, dt = 4t e^{t} - \int 4 e^{t} \, dt = 4t e^{t} - 4 e^{t} \] 5. Combine Results: Substitute the result back into the expression for \( I \): \[ I = 2t^2 e^{t} - (4t e^{t} - 4 e^{t}) + c = 2t^2 e^{t} - 4t e^{t} + 4 e^{t} + c \] \[ I = e^{t} (2t^2 - 4t + 4) + c \] 6. Substitute Back \( t = \sqrt{x} \): Replace \( t \) with \( \sqrt{x} \): \[ I = e^{\sqrt{x}} (2x - 4\sqrt{x} + 4) + c \] Therefore, the solution to the integral is: \[ \boxed{e^{\sqrt{x}} (2x - 4\sqrt{x} + 4) + c} \]