The value of the integral `int_(0)^(pi//2)(1)/(1+(tanx)^(101))dx` is equal to

Options

A. 1
B. `(pi)/(6)`
C. `(pi)/(8)`
D. `(pi)/(4)`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \tan^{101} x} \, dx \), we can use a property of definite integrals.

### Step-by-Step Solution:

1. **Set up the integral**:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \tan^{101} x} \, dx \]

2. **Use the substitution**:
We can use the substitution \( x = \frac{\pi}{2} - t \). Then, \( dx = -dt \). When \( x = 0 \), \( t = \frac{\pi}{2} \) and when \( x = \frac{\pi}{2} \), \( t = 0 \). Thus, the limits of integration will change accordingly:
\[ I = \int_{\frac{\pi}{2}}^{0} \frac{1}{1 + \tan^{101} \left( \frac{\pi}{2} - t \right)} (-dt) = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \cot^{101} t} \, dt \]

3. **Simplify the integrand**:
Recall that \( \cot x = \frac{1}{\tan x} \), so:
\[ \cot^{101} t = \frac{1}{\tan^{101} t} \]
Therefore, we can rewrite the integral:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \frac{1}{\tan^{101} t}} \, dt = \int_{0}^{\frac{\pi}{2}} \frac{\tan^{101} t}{\tan^{101} t + 1} \, dt \]

4. **Combine the two expressions for \( I \)**:
Now we have two expressions for \( I \):
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \tan^{101} x} \, dx \quad \text{and} \quad I = \int_{0}^{\frac{\pi}{2}} \frac{\tan^{101} x}{\tan^{101} x + 1} \, dx \]

5. **Add the two integrals**:
Adding these two expressions gives:
\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{1}{1 + \tan^{101} x} + \frac{\tan^{101} x}{\tan^{101} x + 1} \right) dx \]

6. **Simplify the integrand**:
Notice that:
\[ \frac{1}{1 + \tan^{101} x} + \frac{\tan^{101} x}{\tan^{101} x + 1} = \frac{1 + \tan^{101} x}{1 + \tan^{101} x} = 1 \]
Therefore:
\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} \]

7. **Solve for \( I \)**:
Dividing both sides by 2 gives:
\[ I = \frac{\pi}{4} \]

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