Let `I_1=int_0^1e^(x^2)dx and I_2=int_0^(12)2^(x^2)e^(x^2)dx` then the value of `I_1 +I_2` is equal to

Options

A. 1
B. 2
C. e
D. `e^2`

Answer

To solve the problem, we need to evaluate the integrals \( I_1 \) and \( I_2 \) and find \( I_1 + I_2 \).

### Step 1: Define the integrals
Let:
\[ I_1 = \int_0^1 e^{x^2} \, dx \]
\[ I_2 = \int_0^{1} 2^{x^2} e^{x^2} \, dx \]

### Step 2: Rewrite \( I_2 \)
We can rewrite \( I_2 \) as:
\[ I_2 = \int_0^{1} 2^{x^2} e^{x^2} \, dx = \int_0^{1} e^{x^2} \cdot 2^{x^2} \, dx \]

### Step 3: Use integration by parts on \( I_2 \)
We will perform integration by parts on \( I_2 \). Let:
- \( u = x \)
- \( dv = 2x e^{x^2} \, dx \)

Then:
- \( du = dx \)
- \( v = e^{x^2} \)

Using integration by parts:
\[ I_2 = \left[ x e^{x^2} \right]_0^1 - \int_0^1 e^{x^2} \, dx \]

### Step 4: Evaluate the boundary term
Now, we evaluate the boundary term:
\[ \left[ x e^{x^2} \right]_0^1 = 1 \cdot e^{1^2} - 0 \cdot e^{0^2} = e - 0 = e \]

### Step 5: Substitute back into the equation for \( I_2 \)
Now substituting back into the equation for \( I_2 \):
\[ I_2 = e - I_1 \]

### Step 6: Find \( I_1 + I_2 \)
Now, we can find \( I_1 + I_2 \):
\[ I_1 + I_2 = I_1 + (e - I_1) = e \]

### Final Result
Thus, the value of \( I_1 + I_2 \) is:
\[ \boxed{e} \]