`int(cosx)/((1+sin^(2)x))dx`

Answer

To solve the integral

\[ \int \frac{\cos x}{1 + \sin^2 x} \, dx, \]

we can use a substitution method. Let's go through the steps:

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

### Step 2: Substitute in the integral
Now, substituting \( t \) into the integral, we have:
\[ \int \frac{\cos x}{1 + \sin^2 x} \, dx = \int \frac{\cos x}{1 + t^2} \cdot \frac{dt}{\cos x}. \]
The \( \cos x \) terms cancel out:
\[ = \int \frac{1}{1 + t^2} \, dt. \]

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

### Step 4: Substitute back
Now, substituting back \( t = \sin x \):
\[ = \tan^{-1}(\sin x) + C. \]

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