`intcossqrtxdx =`

Options

A. `2[sqrtxsinsqrtx +cossqrtx]`
B. `sqrtx sinsqrtx - cos sqrt x `
C. `2[sqrtxsinsqrtx - cos sqrtx]`
D. None of these

Answer

To solve the integral \( \int \cos(\sqrt{x}) \, dx \), 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:
\[ x = t^2 \quad \text{and} \quad dx = 2t \, dt \]

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

### Step 3: Integration by Parts
Now we will use integration by parts. The formula for integration by parts is:
\[ \int u \, dv = uv - \int v \, du \]
Here, we can choose:
- \( u = t \) (which implies \( du = dt \))
- \( dv = \cos(t) \, dt \) (which implies \( v = \sin(t) \))

### Step 4: Apply Integration by Parts
Applying the integration by parts formula:
\[ 2 \int t \cos(t) \, dt = 2 \left( t \sin(t) - \int \sin(t) \, dt \right) \]
Now, we need to evaluate \( \int \sin(t) \, dt \):
\[ \int \sin(t) \, dt = -\cos(t) \]
So, substituting back, we have:
\[ 2 \left( t \sin(t) + \cos(t) \right) \]

### Step 5: Substitute Back to Original Variable
Now, substituting \( t = \sqrt{x} \) back into the equation:
\[ 2 \left( \sqrt{x} \sin(\sqrt{x}) + \cos(\sqrt{x}) \right) \]

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

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