`int (cos sqrt(x))/(sqrt(x)) dx` =

Options

A. `2 cos sqrt(x)`
B. `sqrt{(cos x)//x}`
C. `sin sqrt(x)`
D. `2sin sqrt(x)`

Answer

To solve the integral \( \int \frac{\cos(\sqrt{x})}{\sqrt{x}} \, dx \), we will use the substitution method. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = \sqrt{x} \). Then, we have:
\[ x = t^2 \]
Differentiating both sides with respect to \( x \):
\[ dx = 2t \, dt \]

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

### Step 3: Integrate
Now we can integrate:
\[ \int 2 \cos(t) \, dt = 2 \sin(t) + C \]

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

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

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