`int_(0)^((pi)/2)(tan^(7)x)/(cot^(7)x+tan^(7)x)dx` is equal to

Options

A. `(pi)/2`
B. `(pi)/4`
C. `(pi)/6`
D. `(pi)/3`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{\tan^7 x}{\cot^7 x + \tan^7 x} \, dx \), we can use a property of definite integrals. Here’s a step-by-step solution:

### Step 1: Set up the integral
Let
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\tan^7 x}{\cot^7 x + \tan^7 x} \, dx \]

### Step 2: Use the property of definite integrals
We can use the substitution \( x = \frac{\pi}{2} - t \). Then, \( dx = -dt \) and the limits change as follows:
- When \( x = 0 \), \( t = \frac{\pi}{2} \)
- When \( x = \frac{\pi}{2} \), \( t = 0 \)

Thus, we have:
\[ I = \int_{\frac{\pi}{2}}^{0} \frac{\tan^7\left(\frac{\pi}{2} - t\right)}{\cot^7\left(\frac{\pi}{2} - t\right) + \tan^7\left(\frac{\pi}{2} - t\right)} (-dt) \]
This simplifies to:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cot^7 t}{\tan^7 t + \cot^7 t} \, dt \]

### Step 3: Combine the two integrals
Now we have:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\tan^7 x}{\cot^7 x + \tan^7 x} \, dx \]
and
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cot^7 x}{\tan^7 x + \cot^7 x} \, dx \]

Adding these two expressions for \( I \):
\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{\tan^7 x}{\cot^7 x + \tan^7 x} + \frac{\cot^7 x}{\tan^7 x + \cot^7 x} \right) dx \]

### Step 4: Simplify the integrand
The sum simplifies to:
\[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\tan^7 x + \cot^7 x}{\tan^7 x + \cot^7 x} \, dx = \int_{0}^{\frac{\pi}{2}} 1 \, dx \]

### Step 5: Evaluate the integral
Now we can evaluate the integral:
\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]

### Step 6: Solve for \( I \)
Thus, we have:
\[ 2I = \frac{\pi}{2} \implies I = \frac{\pi}{4} \]

### Final Answer
The value of the integral is:
\[ \boxed{\frac{\pi}{4}} \]