`intx^(2)e^(x^(3))cos(e^(x^(3)))dx` is equalto

Options

A. `sin(e^(x^(3)))+C`
B. `3sin(e^(x^(3)))+C`
C. `(1)/(3)sin(e^(x^(3)))+C`
D. `e^(x)sin(e^(x^(3)))+C`

Explanation

To solve the integral \( \int x^2 e^{x^3} \cos(e^{x^3}) \, dx \), we will use substitution. Here are the step-by-step details:

### Step 1: Choose a substitution
Let:
\[ t = e^{x^3} \]
Then, we need to find \( dt \).

### Step 2: Differentiate the substitution
Differentiating both sides with respect to \( x \):
\[ dt = \frac{d}{dx}(e^{x^3}) = e^{x^3} \cdot \frac{d}{dx}(x^3) = e^{x^3} \cdot 3x^2 \, dx \]
This implies:
\[ dx = \frac{dt}{3x^2 e^{x^3}} \]

### Step 3: Substitute in the integral
Now, we can substitute \( t \) and \( dx \) into the integral:
\[ \int x^2 e^{x^3} \cos(e^{x^3}) \, dx = \int x^2 \cdot e^{x^3} \cdot \cos(t) \cdot \frac{dt}{3x^2 e^{x^3}} \]
The \( x^2 e^{x^3} \) terms cancel out:
\[ = \int \frac{1}{3} \cos(t) \, dt \]

### Step 4: Integrate
Now we can integrate:
\[ \int \frac{1}{3} \cos(t) \, dt = \frac{1}{3} \sin(t) + C \]

### Step 5: Substitute back
Now we substitute back \( t = e^{x^3} \):
\[ = \frac{1}{3} \sin(e^{x^3}) + C \]

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