Evaluate ∫ sec x ⋅ tan x ⋅ d x

Answer

To evaluate the integral \( \int \sec x \cdot \tan x \, dx \), we can follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ I = \int \sec x \cdot \tan x \, dx \]

### Step 2: Use substitution
Recall that:
\[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x} \]
Thus, we can rewrite the integral as:
\[ I = \int \frac{\sin x}{\cos^2 x} \, dx \]

### Step 3: Choose a substitution
Let:
\[ t = \cos x \]
Then, differentiating both sides gives:
\[ dt = -\sin x \, dx \quad \Rightarrow \quad \sin x \, dx = -dt \]

### Step 4: Substitute in the integral
Substituting \( t \) into the integral, we have:
\[ I = \int \frac{-dt}{t^2} \]

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

### Step 6: Substitute back for \( t \)
Recalling that \( t = \cos x \), we substitute back:
\[ I = \frac{1}{\cos x} + C \]

### Step 7: Final answer
Since \( \frac{1}{\cos x} = \sec x \), we can write the final answer as:
\[ I = \sec x + C \]