Word Count : 286

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We will solve the integral manually step by step: \[ I = \int \frac{\sqrt{x}}{\sqrt{x} + 2} \,dx \] ### Step 1: Substitution Let: \[ t = \sqrt{x} \Rightarrow dt = \frac{1}{2\sqrt{x}}dx = \frac{1}{2t}dx \] Since \( x = t^2 \), we rewrite the integral in terms of \( t \): \[ dx = 2t \,dt \] Rewriting the integral: \[ I = \int \frac{t}{t + 2} \cdot 2t \, dt \] ________ ___ ___ ____ ________ ___ ______ __________ ___ __________.
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