If `I_(1)=int_(0)^(1) 2^(x^(2)) dx, I_(2)=int_(0)^(1) 2^(x^(3)) dx, I_(3)=int_(1)^(2) 2^(x^(2))dx`
and ` I_(4)=int_(1)^(2) 2^(x^(3))dx` then

Options

A. `I_(1) gt I_(2)` and `I_(4) gt I_(3)`
B. `I_(2)gt I_(1)` and `I_(3) gt I_(4)`
C. `I_(1) gt I_(2)` and `I_(3) gt I_(4)`
D. none of these

Answer

To solve the problem, we need to analyze the integrals \( I_1, I_2, I_3, \) and \( I_4 \) defined as follows:

- \( I_1 = \int_{0}^{1} 2^{x^2} \, dx \)
- \( I_2 = \int_{0}^{1} 2^{x^3} \, dx \)
- \( I_3 = \int_{1}^{2} 2^{x^2} \, dx \)
- \( I_4 = \int_{1}^{2} 2^{x^3} \, dx \)

### Step 1: Compare \( I_1 \) and \( I_2 \)

For \( x \) in the interval \( [0, 1] \):
- We know that \( x^2 \geq x^3 \) since \( x \) is between 0 and 1.
- Therefore, \( 2^{x^2} \geq 2^{x^3} \).

Now, we integrate both sides over the interval \( [0, 1] \):
\[ \int_{0}^{1} 2^{x^2} \, dx \geq \int_{0}^{1} 2^{x^3} \, dx \]
This implies:
\[ I_1 \geq I_2 \]

### Step 2: Compare \( I_3 \) and \( I_4 \)

For \( x \) in the interval \( [1, 2] \):
- We know that \( x^3 \geq x^2 \) since \( x \) is greater than 1.
- Therefore, \( 2^{x^3} \geq 2^{x^2} \).

Now, we integrate both sides over the interval \( [1, 2] \):
\[ \int_{1}^{2} 2^{x^3} \, dx \geq \int_{1}^{2} 2^{x^2} \, dx \]
This implies:
\[ I_4 \geq I_3 \]

### Conclusion

From the above comparisons, we have established:
1. \( I_1 \geq I_2 \)
2. \( I_4 \geq I_3 \)

Thus, the correct relationship is:
\[ I_1 > I_2 \quad \text{and} \quad I_4 > I_3 \]

This corresponds to the first option provided in the question.