Consider `A=int_(0)^(1)(dx)/(1+x^(3))`, then A satisfies

Options

A. `A gt (pi)/(4)`
B. `A lt (pi)/(4)`
C. `A=(pi)/(4)`
D. `A=(pi)/(6)`

Answer

To solve the integral \( A = \int_0^1 \frac{dx}{1+x^3} \) and determine which of the given options it satisfies, we will follow these steps:

### Step 1: Set up the integral
We are given:
\[ A = \int_0^1 \frac{dx}{1+x^3} \]

### Step 2: Establish an inequality
We know that for \( x \) in the interval \([0, 1]\), \( x^3 < x^2 \). Therefore, we can add 1 to both sides:
\[ 1 + x^3 < 1 + x^2 \]
Taking the reciprocal (and reversing the inequality):
\[ \frac{1}{1+x^3} > \frac{1}{1+x^2} \]

### Step 3: Integrate both sides
Now, we can integrate both sides from 0 to 1:
\[ \int_0^1 \frac{dx}{1+x^3} > \int_0^1 \frac{dx}{1+x^2} \]
This gives us:
\[ A > \int_0^1 \frac{dx}{1+x^2} \]

### Step 4: Evaluate the right-hand side integral
The integral \( \int_0^1 \frac{dx}{1+x^2} \) can be computed as follows:
\[ \int_0^1 \frac{dx}{1+x^2} = \tan^{-1}(x) \bigg|_0^1 = \tan^{-1}(1) - \tan^{-1}(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4} \]

### Step 5: Combine results
From the previous steps, we have:
\[ A > \frac{\pi}{4} \]

### Conclusion
Thus, we can conclude that:
\[ A > \frac{\pi}{4} \]
This means the correct option is:
**Option A: \( A > \frac{\pi}{4} \)**.