Evaluate `int sec^(3)x dx`.

Answer

To evaluate the integral \( \int \sec^3 x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting \( \sec^3 x \) as \( \sec^2 x \cdot \sec x \):
\[ \int \sec^3 x \, dx = \int \sec^2 x \cdot \sec x \, dx \]

### Step 2: Use the Identity
We know from trigonometric identities that:
\[ \sec^2 x = 1 + \tan^2 x \]
Thus, we can express \( \sec^2 x \) in terms of \( \tan x \):
\[ \int \sec^3 x \, dx = \int (1 + \tan^2 x) \sec x \, dx \]

### Step 3: Substitution
Let \( t = \tan x \). Then, the derivative \( dt = \sec^2 x \, dx \) implies \( dx = \frac{dt}{\sec^2 x} \). Therefore, we can replace \( \sec^2 x \, dx \) with \( dt \):
\[ \int \sec^3 x \, dx = \int (1 + t^2) \sec x \cdot \frac{dt}{\sec^2 x} = \int (1 + t^2) \sec x \cdot dt \]

### Step 4: Express \( \sec x \)
Using the substitution \( t = \tan x \), we have:
\[ \sec x = \sqrt{1 + \tan^2 x} = \sqrt{1 + t^2} \]
Thus, the integral becomes:
\[ \int (1 + t^2) \sqrt{1 + t^2} \, dt \]

### Step 5: Expand the Integral
Now we expand the integrand:
\[ \int (1 + t^2) \sqrt{1 + t^2} \, dt = \int (1 + t^2)^{3/2} \, dt \]

### Step 6: Use the Formula for Integration
The integral \( \int (1 + t^2)^{3/2} \, dt \) can be solved using the formula:
\[ \int (a^2 + x^2)^{n} \, dx = \frac{x}{2} (a^2 + x^2)^{n} + \frac{a^2}{2n + 1} \int (a^2 + x^2)^{n - 1} \, dx \]
For our case, \( a = 1 \), \( n = \frac{3}{2} \).

### Step 7: Solve the Integral
Using the formula, we can compute:
\[ \int (1 + t^2)^{3/2} \, dt = \frac{t}{2} (1 + t^2)^{3/2} + \frac{1}{2 \cdot \frac{3}{2} + 1} \int (1 + t^2)^{1/2} \, dt \]

### Step 8: Final Result
After evaluating the integral and substituting back \( t = \tan x \), we obtain:
\[ \int \sec^3 x \, dx = \frac{1}{2} \tan x \sec x + \frac{1}{2} \log |\tan x + \sec x| + C \]