`int_(-pi)^(pi) tan x sec^(2) x dx` = _________.

Options

A. 1
B. `-1`
C. 0
D. 2

Answer

To solve the integral \( I = \int_{-\pi}^{\pi} \tan x \sec^2 x \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( y = \tan x \). Then, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \sec^2 x \implies dy = \sec^2 x \, dx \]
This means we can replace \( \sec^2 x \, dx \) with \( dy \).

### Step 2: Change the Limits
Next, we need to change the limits of integration. When \( x = -\pi \):
\[ y = \tan(-\pi) = 0 \]
When \( x = \pi \):
\[ y = \tan(\pi) = 0 \]
Thus, both limits of integration change to 0.

### Step 3: Rewrite the Integral
Now we can rewrite the integral in terms of \( y \):
\[ I = \int_{0}^{0} y \, dy \]

### Step 4: Evaluate the Integral
Since the integral is from 0 to 0, it evaluates to:
\[ I = 0 \]

### Conclusion
Thus, the value of the integral is:
\[ \int_{-\pi}^{\pi} \tan x \sec^2 x \, dx = 0 \]