`intx^3e^(x^2)` dx is equal to

Options

A. `x^2(e^(x^2)-1)`
B. `1/2x^2(e^(x^2)-1)`
C. `1/2e^(x^2)(x^2-1)`
D. `1/2(e^(x^2)-1)`

Answer

To solve the integral \( \int x^3 e^{x^2} \, dx \), we will use substitution and integration by parts. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = x^2 \). Then, differentiate both sides:
\[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x} \]
Now, we can express \( x^3 \) in terms of \( t \):
\[ x^3 = x \cdot x^2 = x \cdot t \]
Thus, the integral becomes:
\[ \int x^3 e^{x^2} \, dx = \int x t e^t \cdot \frac{dt}{2x} \]
The \( x \) in the numerator and denominator cancels out:
\[ = \frac{1}{2} \int t e^t \, dt \]

### Step 2: Integration by Parts
We will use integration by parts on \( \int t e^t \, dt \). Let:
- \( u = t \) \(\Rightarrow du = dt\)
- \( dv = e^t \, dt \) \(\Rightarrow v = e^t\)

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

### Step 3: Substitute Back
Now we substitute back into our integral:
\[ \frac{1}{2} \int t e^t \, dt = \frac{1}{2} \left( t e^t - e^t \right) + C \]
Substituting \( t = x^2 \):
\[ = \frac{1}{2} \left( x^2 e^{x^2} - e^{x^2} \right) + C \]
Factoring out \( e^{x^2} \):
\[ = \frac{1}{2} e^{x^2} (x^2 - 1) + C \]

### Final Answer
Thus, the final answer is:
\[ \int x^3 e^{x^2} \, dx = \frac{1}{2} e^{x^2} (x^2 - 1) + C \]

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