Choose the correct answer `intx^2e^x^3dx` equals (A) `1/3e^x^3+C` (B) `1/3e^x^2+C` (C) `1/2e^x^3+C` (D) `1/2e^x^2+C`

Explanation

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

### Step 1: Choose a substitution
Let \( t = x^3 \). Then, differentiate both sides to find \( dt \):
\[ dt = 3x^2 \, dx \implies dx = \frac{dt}{3x^2} \]

### Step 2: Rewrite \( x^2 \, dx \)
From the substitution, we can express \( x^2 \, dx \) in terms of \( dt \):
\[ x^2 \, dx = \frac{dt}{3} \]

### Step 3: Substitute into the integral
Now, substitute \( t \) and \( dx \) into the integral:
\[ \int x^2 e^{x^3} \, dx = \int e^t \cdot \frac{dt}{3} \]

### Step 4: Factor out the constant
Factor out the constant \( \frac{1}{3} \):
\[ = \frac{1}{3} \int e^t \, dt \]

### Step 5: Integrate \( e^t \)
The integral of \( e^t \) is simply \( e^t \):
\[ = \frac{1}{3} e^t + C \]

### Step 6: Substitute back for \( t \)
Now, substitute back \( t = x^3 \):
\[ = \frac{1}{3} e^{x^3} + C \]

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