Find the value of `int_(0)^(2pi)1/(1+tan^(4)x)dx`

Explanation

To solve the integral \( I = \int_0^{2\pi} \frac{1}{1 + \tan^4 x} \, dx \), we can use properties of definite integrals and symmetry.

### Step 1: Use the property of definite integrals
We know that for any function \( f(x) \):
\[ \int_0^{2a} f(x) \, dx = \int_0^{2a} f(2a - x) \, dx \]
In our case, \( a = \pi \), so we have:
\[ I = \int_0^{2\pi} \frac{1}{1 + \tan^4 x} \, dx = \int_0^{2\pi} \frac{1}{1 + \tan^4(2\pi - x)} \, dx \]

### Step 2: Simplify \( \tan(2\pi - x) \)
Using the property of the tangent function:
\[ \tan(2\pi - x) = -\tan(x) \]
Thus,
\[ \tan^4(2\pi - x) = \tan^4(x) \]
This means:
\[ I = \int_0^{2\pi} \frac{1}{1 + \tan^4 x} \, dx = \int_0^{2\pi} \frac{1}{1 + \tan^4 x} \, dx \]
This confirms that the function is symmetric about \( \pi \).

### Step 3: Split the integral
We can split the integral into two parts:
\[ I = \int_0^{\pi} \frac{1}{1 + \tan^4 x} \, dx + \int_{\pi}^{2\pi} \frac{1}{1 + \tan^4 x} \, dx \]
Using the substitution \( x = 2\pi - t \) in the second integral:
\[ \int_{\pi}^{2\pi} \frac{1}{1 + \tan^4 x} \, dx = \int_0^{\pi} \frac{1}{1 + \tan^4(2\pi - t)} (-dt) = \int_0^{\pi} \frac{1}{1 + \tan^4 t} \, dt \]
Thus:
\[ I = 2 \int_0^{\pi} \frac{1}{1 + \tan^4 x} \, dx \]

### Step 4: Further simplify the integral
Now we can evaluate:
\[ I = 2 \int_0^{\pi} \frac{1}{1 + \tan^4 x} \, dx \]
We can further simplify this integral using the substitution \( x = \frac{\pi}{2} - u \):
\[ \tan\left(\frac{\pi}{2} - u\right) = \cot u \]
Thus:
\[ \tan^4\left(\frac{\pi}{2} - u\right) = \cot^4 u = \frac{1}{\tan^4 u} \]
This gives:
\[ \int_0^{\pi} \frac{1}{1 + \tan^4 x} \, dx = \int_0^{\pi} \frac{\tan^4 u}{\tan^4 u + 1} \, du \]
Adding these two integrals:
\[ 2I = \int_0^{\pi} \left( \frac{1}{1 + \tan^4 x} + \frac{\tan^4 x}{1 + \tan^4 x} \right) dx = \int_0^{\pi} 1 \, dx = \pi \]
Thus:
\[ I = \frac{\pi}{2} \]

### Step 5: Final result
Since we have \( I = 2 \int_0^{\pi} \frac{1}{1 + \tan^4 x} \, dx \), we find that:
\[ I = \pi \]

### Conclusion
The value of the integral is:
\[ \int_0^{2\pi} \frac{1}{1 + \tan^4 x} \, dx = \pi \]