`int sin sqrtx dx`

Answer

To solve the integral \( \int \sin(\sqrt{x}) \, dx \), we will use substitution and integration by parts. Here are the steps:

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

### Step 2: Rewrite the Integral
Substituting \( t \) and \( dx \) into the integral, we get:
\[ \int \sin(\sqrt{x}) \, dx = \int \sin(t) \cdot 2t \, dt = 2 \int t \sin(t) \, dt \]

### Step 3: Integration by Parts
Now, we will use integration by parts. Let:
- \( u = t \) (thus, \( du = dt \))
- \( dv = \sin(t) \, dt \) (thus, \( v = -\cos(t) \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we have:
\[ \int t \sin(t) \, dt = -t \cos(t) - \int -\cos(t) \, dt \]
\[ = -t \cos(t) + \int \cos(t) \, dt \]
\[ = -t \cos(t) + \sin(t) \]

### Step 4: Substitute Back
Now, substituting back into our integral:
\[ 2 \int t \sin(t) \, dt = 2 \left( -t \cos(t) + \sin(t) \right) \]
\[ = -2t \cos(t) + 2\sin(t) \]

### Step 5: Replace \( t \) with \( \sqrt{x} \)
Now, we replace \( t \) back with \( \sqrt{x} \):
\[ = -2\sqrt{x} \cos(\sqrt{x}) + 2\sin(\sqrt{x}) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \sin(\sqrt{x}) \, dx = -2\sqrt{x} \cos(\sqrt{x}) + 2\sin(\sqrt{x}) + C \]

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