`int(sinx)/(sqrt(4+cos^(2)x))dx`

Explanation

To solve the integral \( \int \frac{\sin x}{\sqrt{4 + \cos^2 x}} \, dx \), we will use a substitution method. Here are the steps:

### Step 1: Substitution
Let \( t = \cos x \). Then, the derivative of \( t \) with respect to \( x \) is:
\[ \frac{dt}{dx} = -\sin x \quad \Rightarrow \quad \sin x \, dx = -dt \]

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ \int \frac{\sin x}{\sqrt{4 + \cos^2 x}} \, dx = \int \frac{-dt}{\sqrt{4 + t^2}} \]

### Step 3: Factor Out the Negative Sign
We can factor out the negative sign:
\[ = -\int \frac{dt}{\sqrt{4 + t^2}} \]

### Step 4: Use the Integral Formula
The integral \( \int \frac{dt}{\sqrt{a^2 + t^2}} \) has a known result:
\[ \int \frac{dt}{\sqrt{a^2 + t^2}} = \ln |t + \sqrt{t^2 + a^2}| + C \]
Here, \( a = 2 \). Therefore, we can write:
\[ -\int \frac{dt}{\sqrt{4 + t^2}} = -\left( \ln |t + \sqrt{t^2 + 4}| + C \right) \]

### Step 5: Substitute Back for \( t \)
Now, substituting back \( t = \cos x \):
\[ = -\left( \ln |\cos x + \sqrt{\cos^2 x + 4}| + C \right) \]

### Step 6: Simplify the Expression
This can be simplified to:
\[ = -\ln |\cos x + \sqrt{\cos^2 x + 4}| + C \]

### Final Answer
Thus, the final result for the integral is:
\[ \int \frac{\sin x}{\sqrt{4 + \cos^2 x}} \, dx = -\ln |\cos x + \sqrt{\cos^2 x + 4}| + C \]

---