What is `int(sinsqrt(x))/(sqrt(x))dx` is equal to ?

Options

A. `(cossqrt(x))/(2)+c`
B. `2cossqrt(x)+C`
C. `(-cossqrt(x))/(2)+C`
D. `-2cossqrt(x)+C`

Answer

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

### Step 1: Substitution
Let \( u = \sqrt{x} \). Then, we have:
\[ x = u^2 \]
Differentiating both sides gives:
\[ dx = 2u \, du \]

### Step 2: Rewrite the Integral
Now, we can rewrite the integral in terms of \( u \):
\[ \sqrt{x} = u \quad \text{and} \quad dx = 2u \, du \]
Thus, the integral becomes:
\[ \int \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx = \int \frac{\sin(u)}{u} (2u \, du) \]
This simplifies to:
\[ \int 2 \sin(u) \, du \]

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

### Step 4: Substitute Back
Now we substitute back \( u = \sqrt{x} \):
\[ -2 \cos(\sqrt{x}) + C \]

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

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