The value of definite integral `int _(0)^(oo) (dx )/((1+ x^(9)) (1+ x^(2)))` equal to:

Options

A. `(pi)/(16)`
B. `(pi)/(8)`
C. `(pi)/(4)`
D. `(pi)/(2)`

Answer

To solve the definite integral

\[ I = \int_{0}^{\infty} \frac{dx}{(1 + x^9)(1 + x^2)}, \]

we will use the substitution \( x = \tan \theta \).

### Step 1: Substitute \( x = \tan \theta \)

When \( x = \tan \theta \), we have:

\[ dx = \sec^2 \theta \, d\theta. \]

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

Thus, the integral becomes:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sec^2 \theta \, d\theta}{(1 + \tan^9 \theta)(1 + \tan^2 \theta)}. \]

### Step 2: Simplify the integral

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

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sec^2 \theta \, d\theta}{(1 + \tan^9 \theta) \sec^2 \theta} = \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{1 + \tan^9 \theta}. \]

### Step 3: Use the property of integrals

Next, we will use the property of integrals:

\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx. \]

Here, we have \( a = 0 \) and \( b = \frac{\pi}{2} \), so:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{1 + \tan^9 \theta} = \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{1 + \cot^9 \theta}. \]

### Step 4: Rewrite the second integral

For the second integral, we can express \( \cot \theta \) in terms of \( \tan \theta \):

\[ \cot^9 \theta = \frac{1}{\tan^9 \theta}. \]

Thus, we have:

\[ \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{1 + \cot^9 \theta} = \int_{0}^{\frac{\pi}{2}} \frac{\tan^9 \theta \, d\theta}{\tan^9 \theta + 1}. \]

### Step 5: Combine both integrals

Now we can add both integrals:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{1}{1 + \tan^9 \theta} + \frac{\tan^9 \theta}{\tan^9 \theta + 1} \right) d\theta. \]

The expression simplifies to:

\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, d\theta = \frac{\pi}{2}. \]

### Step 6: Solve for \( I \)

Thus, we find:

\[ I = \frac{\pi}{4}. \]

### Conclusion

The value of the definite integral is:

\[ \int_{0}^{\infty} \frac{dx}{(1 + x^9)(1 + x^2)} = \frac{\pi}{4}. \]