`int(sinx)/((1+cos^(2)x))dx`

Explanation

To solve the integral \( \int \frac{\sin x}{1 + \cos^2 x} \, dx \), we can use a substitution method. Here’s a step-by-step solution:

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

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we get:
\[ \int \frac{\sin x}{1 + \cos^2 x} \, dx = \int \frac{-dt}{1 + t^2} \]

### Step 3: Integrate
The integral \( \int \frac{-dt}{1 + t^2} \) is a standard integral that evaluates to:
\[ -\tan^{-1}(t) + C \]

### Step 4: Substitute Back
Now, we substitute back \( t = \cos x \):
\[ -\tan^{-1}(\cos x) + C \]

### Final Answer
Thus, the integral evaluates to:
\[ \int \frac{\sin x}{1 + \cos^2 x} \, dx = -\tan^{-1}(\cos x) + C \]