The value of `int_(0)^(oo)(dx)/(1+x^(4))` is equal to

Options

A. `(pi)/(2sqrt2)`
B. `(pi)/(2)`
C. `(pi)/(sqrt2)`
D. `2pisqrt2`

Answer

To solve the integral \( I = \int_0^\infty \frac{dx}{1+x^4} \), we can use a symmetry property and some substitutions. Here’s a step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int_0^\infty \frac{dx}{1+x^4} \]

### Step 2: Use a Substitution
To evaluate this integral, we can use the substitution \( x = \frac{1}{t} \). Then, \( dx = -\frac{1}{t^2} dt \). The limits change as follows:
- When \( x = 0 \), \( t \to \infty \)
- When \( x \to \infty \), \( t = 0 \)

Thus, the integral becomes:
\[ I = \int_\infty^0 \frac{-\frac{1}{t^2}}{1+\left(\frac{1}{t}\right)^4} dt = \int_0^\infty \frac{dt}{t^2 \left(1 + \frac{1}{t^4}\right)} = \int_0^\infty \frac{dt}{\frac{t^4 + 1}{t^2}} = \int_0^\infty \frac{t^2}{t^4 + 1} dt \]

### Step 3: Combine the Integrals
Now we have two expressions for \( I \):
1. \( I = \int_0^\infty \frac{dx}{1+x^4} \)
2. \( I = \int_0^\infty \frac{x^2}{1+x^4} dx \)

Adding these two integrals gives:
\[ 2I = \int_0^\infty \left( \frac{1+x^2}{1+x^4} \right) dx \]

### Step 4: Simplify the Integral
Now, simplify the integrand:
\[ \frac{1+x^2}{1+x^4} = \frac{1}{1+x^4} + \frac{x^2}{1+x^4} \]
Thus,
\[ 2I = \int_0^\infty \frac{dx}{1+x^4} + \int_0^\infty \frac{x^2}{1+x^4} dx \]

### Step 5: Evaluate the Integral
Notice that:
\[ \int_0^\infty \frac{x^2}{1+x^4} dx = \int_0^\infty \frac{u}{1+u^2} \cdot \frac{du}{2} \quad \text{(where \( u = x^2 \))} \]
This integral can be evaluated using the substitution \( u = x^2 \), leading to:
\[ \int_0^\infty \frac{u}{1+u^2} du = \frac{\pi}{2} \]
Thus,
\[ \int_0^\infty \frac{x^2}{1+x^4} dx = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4} \]

### Step 6: Final Calculation
Now we can substitute back:
\[ 2I = I + \frac{\pi}{4} \]
This gives:
\[ I = \frac{\pi}{4} \]
Thus, the final value of the integral is:
\[ I = \frac{\pi}{4} \]

### Conclusion
The value of the integral \( \int_0^\infty \frac{dx}{1+x^4} \) is:
\[ \boxed{\frac{\pi}{4}} \]