`intsinsqrt(x) dx`

Answer

To solve the integral \(\int \sin(\sqrt{x}) \, dx\), we can follow these steps:

### Step 1: Substitution
Let \( T = \sqrt{x} \). Then, we differentiate both sides:
\[ \frac{dT}{dx} = \frac{1}{2\sqrt{x}} \implies dx = 2T \, dT \]

### Step 2: Rewrite the Integral
Substituting \( T \) and \( dx \) into the integral gives:
\[ \int \sin(\sqrt{x}) \, dx = \int \sin(T) \cdot 2T \, dT = 2 \int T \sin(T) \, dT \]

### Step 3: Integration by Parts
Now we will use integration by parts. We let:
- \( u = T \) (first part)
- \( dv = \sin(T) \, dT \) (second part)

Then, we differentiate and integrate:
- \( du = dT \)
- \( v = -\cos(T) \)

Using the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
we have:
\[ 2 \int T \sin(T) \, dT = 2 \left( -T \cos(T) - \int -\cos(T) \, dT \right) \]

### Step 4: Simplifying the Integral
Now we simplify:
\[ = 2 \left( -T \cos(T) + \int \cos(T) \, dT \right) \]
The integral of \(\cos(T)\) is \(\sin(T)\):
\[ = 2 \left( -T \cos(T) + \sin(T) \right) \]

### Step 5: Substitute Back
Now we substitute back \( T = \sqrt{x} \):
\[ = 2 \left( -\sqrt{x} \cos(\sqrt{x}) + \sin(\sqrt{x}) \right) + C \]

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

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