Evaluate `int tan ^(2) x sec^(4) x dx `

Answer

To evaluate the integral \( \int \tan^2 x \sec^4 x \, dx \), we can follow these steps:

### Step 1: Rewrite the integral
We know that \( \sec^4 x = \sec^2 x \cdot \sec^2 x \). Therefore, we can rewrite the integral as:
\[ \int \tan^2 x \sec^4 x \, dx = \int \tan^2 x \sec^2 x \cdot \sec^2 x \, dx \]

### Step 2: Use the identity for \( \sec^2 x \)
Recall the identity:
\[ \sec^2 x = 1 + \tan^2 x \]
This allows us to express the integral in terms of \( \tan x \):
\[ \int \tan^2 x \sec^4 x \, dx = \int \tan^2 x (1 + \tan^2 x) \sec^2 x \, dx \]

### Step 3: Substitute \( t = \tan x \)
Let \( t = \tan x \). Then, the derivative \( dt = \sec^2 x \, dx \). Thus, we can rewrite the integral:
\[ \int \tan^2 x \sec^4 x \, dx = \int t^2 (1 + t^2) \, dt \]

### Step 4: Expand the integrand
Now, expand the integrand:
\[ \int t^2 (1 + t^2) \, dt = \int (t^2 + t^4) \, dt \]

### Step 5: Integrate term by term
Now, we can integrate each term separately:
\[ \int t^2 \, dt = \frac{t^3}{3} \quad \text{and} \quad \int t^4 \, dt = \frac{t^5}{5} \]
Thus,
\[ \int (t^2 + t^4) \, dt = \frac{t^3}{3} + \frac{t^5}{5} + C \]

### Step 6: Substitute back \( t = \tan x \)
Finally, substitute back \( t = \tan x \):
\[ \frac{\tan^3 x}{3} + \frac{\tan^5 x}{5} + C \]

### Final Answer
The evaluated integral is:
\[ \int \tan^2 x \sec^4 x \, dx = \frac{\tan^3 x}{3} + \frac{\tan^5 x}{5} + C \]