Evaluate: `intcos^3sqrt(x)\ \ dx`

Explanation

To evaluate the integral \( I = \int \cos^3(\sqrt{x}) \, dx \), we will use substitution and integration by parts. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = \sqrt{x} \). Then, we have:
\[ x = t^2 \quad \text{and} \quad dx = 2t \, dt \]

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