`If I_1=int_0^1 2^x^2,I_2=int_0^1 2^x^3dx ,I_3=int_1^2^x^2dx ,I_4=int_1^2 2^x^3dx ,` then which of the following is/are ture?
a) `I_1> I_2`
(b) `I_2> I_2`
c)`I_3> I_4`
(d) `I_3 < I_4`

Options

A. `I_(1)gtI_(2)`
B. `I_(2)gtI_(1)`
C. `I_(3)gtI_(4)`
D. `I_(3)ltI_(4)`

Explanation

To solve the problem, we need to evaluate the integrals \( I_1, I_2, I_3, \) and \( I_4 \) and compare them based on the given conditions.

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

We have:
\[ I_1 = \int_0^1 2^{x^2} \, dx \]
\[ I_2 = \int_0^1 2^{x^3} \, dx \]

**Comparison of \( I_1 \) and \( I_2 \)**:
For \( x \) in the interval \( [0, 1] \):
- Since \( x^2 > x^3 \) for \( 0 < x < 1 \), we have \( 2^{x^2} > 2^{x^3} \).

Thus, we can conclude:
\[ I_1 > I_2 \]

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

Next, we consider:
\[ I_3 = \int_1^2 2^{x^2} \, dx \]
\[ I_4 = \int_1^2 2^{x^3} \, dx \]

**Comparison of \( I_3 \) and \( I_4 \)**:
For \( x \) in the interval \( [1, 2] \):
- Since \( x^2 < x^3 \) for \( x > 1 \), we have \( 2^{x^2} < 2^{x^3} \).

Thus, we can conclude:
\[ I_3 < I_4 \]

### Summary of Results
From the evaluations:
- \( I_1 > I_2 \) (True)
- \( I_3 < I_4 \) (True)

### Conclusion
The correct options are:
- (a) \( I_1 > I_2 \) is true.
- (d) \( I_3 < I_4 \) is true.