If the integral `I=inte^(x^(2))x^(3)dx=e^(x^(2))f(x)+c`, where c is the constant of integration and `f(1)=0`, then the value of `f(2)` is equal to

Options

A. `4`
B. `(5)/(2)`
C. `(3)/(2)`
D. `3`

Answer

To solve the problem, we need to evaluate the integral \( I = \int e^{x^2} x^3 \, dx \) and express it in the form \( I = e^{x^2} f(x) + c \), where \( c \) is the constant of integration. We also know that \( f(1) = 0 \) and we need to find \( f(2) \).

### Step-by-Step Solution:

1. **Set Up the Integral:**
\[ I = \int e^{x^2} x^3 \, dx \]

2. **Substitution:**
Let \( t = x^2 \). Then, differentiating both sides gives:
\[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x} \]
Since \( x = \sqrt{t} \), we can substitute \( dx \):
\[ dx = \frac{dt}{2\sqrt{t}} \]
Now, substituting in the integral:
\[ I = \int e^{t} (\sqrt{t})^3 \cdot \frac{dt}{2\sqrt{t}} = \int e^{t} \frac{t^{3/2}}{2\sqrt{t}} \, dt = \int e^{t} \frac{t}{2} \, dt \]

3. **Simplifying the Integral:**
\[ I = \frac{1}{2} \int t e^{t} \, dt \]

4. **Integration by Parts:**
For \( \int t e^{t} \, dt \), let:
- \( u = t \) and \( dv = e^{t} dt \)
- Then, \( du = dt \) and \( v = e^{t} \)
Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
We get:
\[ \int t e^{t} \, dt = t e^{t} - \int e^{t} \, dt = t e^{t} - e^{t} + C \]

5. **Substituting Back:**
Now substituting back into the expression for \( I \):
\[ I = \frac{1}{2} \left( t e^{t} - e^{t} \right) + C = \frac{1}{2} e^{t} (t - 1) + C \]
Replacing \( t \) with \( x^2 \):
\[ I = \frac{1}{2} e^{x^2} (x^2 - 1) + C \]

6. **Comparing with Given Form:**
We have:
\[ I = e^{x^2} f(x) + C \]
Thus, comparing:
\[ f(x) = \frac{1}{2} (x^2 - 1) \]

7. **Finding \( f(1) \):**
Check \( f(1) \):
\[ f(1) = \frac{1}{2} (1^2 - 1) = \frac{1}{2} (0) = 0 \]
This satisfies the condition \( f(1) = 0 \).

8. **Finding \( f(2) \):**
Now calculate \( f(2) \):
\[ f(2) = \frac{1}{2} (2^2 - 1) = \frac{1}{2} (4 - 1) = \frac{1}{2} (3) = \frac{3}{2} \]

### Final Answer:
\[ f(2) = \frac{3}{2} \]