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 integral \( I = \int e^{x^2} x^3 \, dx \) and express it in the form \( e^{x^2} f(x) + c \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int e^{x^2} x^3 \, dx \]

### Step 2: Use Substitution
Let \( t = x^2 \). Then, the differential \( dt = 2x \, dx \) or \( dx = \frac{dt}{2x} \). Since \( x = \sqrt{t} \), we can rewrite \( dx \) as:
\[ dx = \frac{dt}{2\sqrt{t}} \]
Now, substituting \( t \) into the integral:
\[ I = \int e^t x^3 \cdot \frac{dt}{2\sqrt{t}} = \int e^t \cdot t^{3/2} \cdot \frac{dt}{2\sqrt{t}} = \frac{1}{2} \int e^t t \, dt \]

### Step 3: Integrate by Parts
We will use integration by parts on \( \int e^t t \, dt \). Let:
- \( u = t \) and \( dv = e^t dt \)
Then, we have:
- \( du = dt \) and \( v = e^t \)

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ \int e^t t \, dt = t e^t - \int e^t \, dt = t e^t - e^t + C \]
Thus,
\[ \int e^t t \, dt = e^t (t - 1) + C \]

### Step 4: Substitute Back
Now substituting back into our integral:
\[ I = \frac{1}{2} \left( e^t (t - 1) + C \right) = \frac{1}{2} e^{x^2} (x^2 - 1) + C \]

### Step 5: Express in Required Form
We can express this as:
\[ I = e^{x^2} \left( \frac{x^2 - 1}{2} \right) + C \]
From this, we identify:
\[ f(x) = \frac{x^2 - 1}{2} \]

### Step 6: Evaluate \( f(2) \)
Now, we need to find \( f(2) \):
\[ f(2) = \frac{2^2 - 1}{2} = \frac{4 - 1}{2} = \frac{3}{2} \]

Thus, the value of \( f(2) \) is:
\[ \boxed{\frac{3}{2}} \]