If `x=int_(0)^(oo)(dt)/((1+t^(2))(1+t^(2017)))`, then `(3x)/(pi)` is equal to

Explanation

To solve the problem, we need to evaluate the integral

\[ x = \int_{0}^{\infty} \frac{dt}{(1+t^2)(1+t^{2017})} \]

and then find the value of \(\frac{3x}{\pi}\).

### Step 1: Change of Variables
We will use the substitution \( t = \tan(\theta) \). This gives us \( dt = \sec^2(\theta) d\theta \).

### Step 2: Adjust the Limits
When \( t = 0 \), \( \theta = 0 \) and when \( t \to \infty \), \( \theta \to \frac{\pi}{2} \).

### Step 3: Rewrite the Integral
Substituting \( t = \tan(\theta) \) into the integral, we have:

\[ x = \int_{0}^{\frac{\pi}{2}} \frac{\sec^2(\theta) d\theta}{(1+\tan^2(\theta))(1+\tan^{2017}(\theta))} \]

### Step 4: Simplify the Integral
Using the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \), we can simplify the integral:

\[ x = \int_{0}^{\frac{\pi}{2}} \frac{\sec^2(\theta) d\theta}{\sec^2(\theta)(1+\tan^{2017}(\theta))} = \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{1+\tan^{2017}(\theta)} \]

### Step 5: Evaluate the Integral
The integral

\[ \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{1+\tan^{2017}(\theta)} \]

is known to evaluate to \(\frac{\pi}{4}\) for any positive integer \( n \) in the form \( \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{1+\tan^n(\theta)} = \frac{\pi}{4}\).

Thus, we have:

\[ x = \frac{\pi}{4} \]

### Step 6: Calculate \(\frac{3x}{\pi}\)
Now we can calculate:

\[ \frac{3x}{\pi} = \frac{3 \cdot \frac{\pi}{4}}{\pi} = \frac{3}{4} \]

### Final Answer
Therefore, the value of \(\frac{3x}{\pi}\) is

\[ \frac{3}{4} \]