`int(cos sqrt(x))/( sqrt(x)) dx ` is equal to

Options

A. ` 2 cos sqrt(x) + C`
B. `2 sin sqrt(x) + C`
C. ` - 2 sin sqrt( x) + C`
D. ` sin sqrt( x) + C`

Answer

To solve the integral \( \int \frac{\cos(\sqrt{x})}{\sqrt{x}} \, dx \), we will use a substitution method. Here are the steps:

### Step 1: Substitution
Let \( t = \sqrt{x} \). Then, we can express \( x \) in terms of \( t \):
\[ x = t^2 \]
Now, we need to find \( dx \) in terms of \( dt \):
\[ dx = 2t \, dt \]

### Step 2: Rewrite the Integral
Now, we can rewrite the integral in terms of \( t \):
\[ \sqrt{x} = t \quad \text{and} \quad dx = 2t \, dt \]
Substituting these into the integral gives:
\[ \int \frac{\cos(\sqrt{x})}{\sqrt{x}} \, dx = \int \frac{\cos(t)}{t} (2t \, dt) \]
This simplifies to:
\[ \int 2 \cos(t) \, dt \]

### Step 3: Integrate
Now, we can integrate:
\[ \int 2 \cos(t) \, dt = 2 \sin(t) + C \]
where \( C \) is the constant of integration.

### Step 4: Back Substitute
Now, we substitute back \( t = \sqrt{x} \):
\[ 2 \sin(t) + C = 2 \sin(\sqrt{x}) + C \]

### Final Answer
Thus, the integral \( \int \frac{\cos(\sqrt{x})}{\sqrt{x}} \, dx \) is equal to:
\[ \boxed{2 \sin(\sqrt{x}) + C} \]

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