`int(f'(x))/(sqrt(f(x)))dx=…+C,f(x) ne 0`

Options

A. `(1)/(2)sqrt(f(x))`
B. `2sqrt(f(x))`
C. `(1)/(2)f(x)`
D. `2f(x)`

Explanation

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

### Step 1: Substitution
Let \(t = f(x)\). Then, the derivative \(dt\) can be expressed in terms of \(f'(x)\):
\[ dt = f'(x) \, dx \quad \Rightarrow \quad dx = \frac{dt}{f'(x)} \]

### Step 2: Rewrite the Integral
Substituting \(t\) into the integral gives us:
\[ \int \frac{f'(x)}{\sqrt{f(x)}} \, dx = \int \frac{f'(x)}{\sqrt{t}} \cdot \frac{dt}{f'(x)} = \int \frac{1}{\sqrt{t}} \, dt \]

### Step 3: Integrate
Now we can integrate \(\frac{1}{\sqrt{t}}\):
\[ \int \frac{1}{\sqrt{t}} \, dt = 2\sqrt{t} + C \]

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

### Final Result
Thus, we have:
\[ \int \frac{f'(x)}{\sqrt{f(x)}} \, dx = 2\sqrt{f(x)} + C \]

### Conclusion
Comparing this with the original question, we find that the expression that comes before \(+ C\) is \(2\sqrt{f(x)}\).