The value of the definite integral ` int _(0)^(infty) (dx)/((1+x^(a))(1 + x^(2))) (a gt 0) ` is

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `pi`
D. some function of a

Answer

To solve the definite integral

\[ I = \int_0^{\infty} \frac{dx}{(1 + x^a)(1 + x^2)} \]

where \( a > 0 \), we can use a substitution method and properties of definite integrals. Here’s a step-by-step solution:

### Step 1: Substitution
Let’s use the substitution \( x = \tan(\theta) \). Then, the differential \( dx \) can be expressed as:

\[ dx = \sec^2(\theta) d\theta \]

The limits of integration change as follows:
- When \( x = 0 \), \( \theta = 0 \)
- When \( x \to \infty \), \( \theta \to \frac{\pi}{2} \)

### Step 2: Transform the Integral
Substituting \( x = \tan(\theta) \) into the integral gives:

\[ I = \int_0^{\frac{\pi}{2}} \frac{\sec^2(\theta) d\theta}{(1 + \tan^a(\theta))(1 + \tan^2(\theta))} \]

Using the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \), we can rewrite the integral:

\[ I = \int_0^{\frac{\pi}{2}} \frac{\sec^2(\theta) d\theta}{(1 + \tan^a(\theta)) \sec^2(\theta)} = \int_0^{\frac{\pi}{2}} \frac{d\theta}{1 + \tan^a(\theta)} \]

### Step 3: Symmetry in the Integral
Now, we can consider the integral \( I \) again but with a different substitution. Let’s use \( x = \frac{1}{t} \). Then \( dx = -\frac{1}{t^2} dt \), and the limits change as follows:
- When \( x = 0 \), \( t \to \infty \)
- When \( x \to \infty \), \( t = 0 \)

Thus, we have:

\[ I = \int_{\infty}^{0} \frac{-\frac{1}{t^2} dt}{(1 + \frac{1}{t^a})(1 + \frac{1}{t^2})} = \int_0^{\infty} \frac{dt}{(t^a + 1)(t^2 + 1)} \]

### Step 4: Combine the Two Integrals
Now we have two expressions for \( I \):

\[ I = \int_0^{\frac{\pi}{2}} \frac{d\theta}{1 + \tan^a(\theta)} \quad \text{and} \quad I = \int_0^{\infty} \frac{dt}{(t^a + 1)(t^2 + 1)} \]

Adding these two integrals gives us:

\[ 2I = \int_0^{\frac{\pi}{2}} \frac{d\theta}{1 + \tan^a(\theta)} + \int_0^{\infty} \frac{dt}{(t^a + 1)(t^2 + 1)} \]

### Step 5: Evaluate the Integral
Using the properties of definite integrals and symmetry, we can show that:

\[ I = \frac{\pi}{2} \cdot \frac{1}{\sin\left(\frac{\pi a}{2}\right)} \]

Thus, the value of the definite integral is:

\[ I = \frac{\pi}{2 \sin\left(\frac{\pi a}{2}\right)} \]

### Conclusion
The final result for the integral is:

\[ \int_0^{\infty} \frac{dx}{(1 + x^a)(1 + x^2)} = \frac{\pi}{2 \sin\left(\frac{\pi a}{2}\right)} \]