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

Answer

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

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int x^3 e^{x^2} \, dx \]
We can factor out a \( \frac{1}{2} \) to make the substitution easier:
\[ = \frac{1}{2} \int 2x^3 e^{x^2} \, dx \]

### Step 2: Substitution
Let \( t = x^2 \). Then, differentiating both sides gives us:
\[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x} \]
Since \( x = \sqrt{t} \), we can express \( dx \) in terms of \( t \):
\[ dx = \frac{dt}{2\sqrt{t}} \]
Now, substituting \( x^3 = (x^2)^{3/2} = t^{3/2} \) and \( 2x \, dx = dt \):
\[ \int x^3 e^{x^2} \, dx = \frac{1}{2} \int t^{3/2} e^t \cdot \frac{dt}{2\sqrt{t}} = \frac{1}{4} \int t e^t \, dt \]

### Step 3: Integration by Parts
Now we will use integration by parts on \( \int t e^t \, dt \). Let:
- \( u = t \) → \( du = dt \)
- \( dv = e^t dt \) → \( v = e^t \)

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

### Step 4: Substitute Back
Now substituting back into our integral:
\[ \int x^3 e^{x^2} \, dx = \frac{1}{4} \left( t e^t - e^t \right) + C \]
Replacing \( t \) back with \( x^2 \):
\[ = \frac{1}{4} \left( x^2 e^{x^2} - e^{x^2} \right) + C \]
Thus, the final answer is:
\[ = \frac{1}{4} e^{x^2} (x^2 - 1) + C \]

### Final Answer
\[ \int x^3 e^{x^2} \, dx = \frac{1}{4} e^{x^2} (x^2 - 1) + C \]