`int(2cos x)/(3sin ^(2)x)dx,`

Answer

To solve the integral \(\int \frac{2 \cos x}{3 \sin^2 x} \, dx\), we will use the substitution method. Here’s a step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \frac{2 \cos x}{3 \sin^2 x} \, dx \]

### Step 2: Use Substitution
Let’s use the substitution \(t = \sin x\). Then, the derivative of \(t\) with respect to \(x\) is:
\[ dt = \cos x \, dx \quad \Rightarrow \quad dx = \frac{dt}{\cos x} \]

### Step 3: Substitute in the Integral
Now, we can substitute \(t\) and \(dx\) into the integral:
\[ \int \frac{2 \cos x}{3 \sin^2 x} \, dx = \int \frac{2 \cos x}{3 t^2} \cdot \frac{dt}{\cos x} \]

### Step 4: Simplify the Integral
The \(\cos x\) terms cancel out:
\[ = \int \frac{2}{3 t^2} \, dt \]

### Step 5: Integrate
Now we can integrate:
\[ \int \frac{2}{3 t^2} \, dt = \frac{2}{3} \int t^{-2} \, dt = \frac{2}{3} \left(-\frac{1}{t}\right) + C = -\frac{2}{3t} + C \]

### Step 6: Substitute Back
Now we substitute back \(t = \sin x\):
\[ -\frac{2}{3 \sin x} + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{2 \cos x}{3 \sin^2 x} \, dx = -\frac{2}{3 \sin x} + C \]

---