The area bounded by `y = tan x, y = cot x,` X-axis in `0 lt=x lt= pi/2` is

Options

A. log 2
B. `(1)/(2)log 2`
C. `2 " log "((1)/(sqrt(2)))`
D. `(3)/(2)log 2`

Answer

To find the area bounded by the curves \( y = \tan x \), \( y = \cot x \), and the x-axis in the interval \( 0 < x < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Identify the points of intersection
To find the area between the curves, we first need to determine where they intersect. We set \( \tan x = \cot x \).

\[ \tan x = \cot x \implies \tan^2 x = 1 \implies x = \frac{\pi}{4} \]

Thus, the curves intersect at \( x = \frac{\pi}{4} \).

### Step 2: Set up the integral for the area
The area \( A \) between the curves from \( x = 0 \) to \( x = \frac{\pi}{4} \) can be calculated by integrating the difference between the upper curve and the lower curve. In this case, \( y = \tan x \) is above \( y = \cot x \) in this interval.

\[ A = \int_0^{\frac{\pi}{4}} (\tan x - \cot x) \, dx \]

### Step 3: Simplify the integrand
We can rewrite \( \cot x \) as \( \frac{1}{\tan x} \). Thus, the integrand becomes:

\[ \tan x - \cot x = \tan x - \frac{1}{\tan x} = \frac{\tan^2 x - 1}{\tan x} \]

### Step 4: Calculate the integral
Now we can compute the integral:

\[ A = \int_0^{\frac{\pi}{4}} \left( \tan x - \cot x \right) \, dx = \int_0^{\frac{\pi}{4}} \left( \tan x - \frac{1}{\tan x} \right) \, dx \]

We can integrate \( \tan x \) and \( \cot x \) separately:

\[ \int \tan x \, dx = -\log(\cos x) + C \]
\[ \int \cot x \, dx = \log(\sin x) + C \]

Thus, we have:

\[ A = \left[-\log(\cos x) - \log(\sin x)\right]_0^{\frac{\pi}{4}} \]

### Step 5: Evaluate the definite integral
Now we evaluate the integral at the limits:

1. At \( x = \frac{\pi}{4} \):
\[ -\log(\cos(\frac{\pi}{4})) - \log(\sin(\frac{\pi}{4})) = -\log\left(\frac{1}{\sqrt{2}}\right) - \log\left(\frac{1}{\sqrt{2}}\right) = -(-\frac{1}{2} \log(2)) - (-\frac{1}{2} \log(2)) = \log(2) \]

2. At \( x = 0 \):
\[ -\log(\cos(0)) - \log(\sin(0)) = -\log(1) - \log(0) \text{ (undefined)} \]

However, we note that as \( x \) approaches \( 0 \), \( \sin x \) approaches \( 0 \), leading to \( -\log(0) \) which diverges negatively.

### Step 6: Combine results
The area from \( 0 \) to \( \frac{\pi}{4} \) is \( \log(2) \). Since the area from \( \frac{\pi}{4} \) to \( \frac{\pi}{2} \) is symmetric, we double the area:

\[ \text{Total Area} = 2 \cdot \log(2) = \log(2^2) = \log(4) \]

### Final Answer
The area bounded by \( y = \tan x \), \( y = \cot x \), and the x-axis in the interval \( 0 < x < \frac{\pi}{2} \) is:

\[ \text{Area} = \log(4) \]