Evaluate: `intx^(2)e^(x^(3))" cos"(2e^(x^(3)))dx`

Answer

To evaluate the integral \( \int x^2 e^{x^3} \cos(2 e^{x^3}) \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = e^{x^3} \). Then, we need to find \( dt \):
\[ dt = \frac{d}{dx}(e^{x^3}) \, dx = e^{x^3} \cdot 3x^2 \, dx \]
This implies:
\[ dx = \frac{dt}{3x^2 e^{x^3}} \]

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

### Step 3: Factor Out the Constant
We can factor out the constant \( \frac{1}{3} \):
\[ = \frac{1}{3} \int \cos(2t) \, dt \]

### Step 4: Integrate
The integral of \( \cos(2t) \) is:
\[ \int \cos(2t) \, dt = \frac{1}{2} \sin(2t) + C \]
Thus, substituting this back, we have:
\[ = \frac{1}{3} \cdot \frac{1}{2} \sin(2t) + C = \frac{1}{6} \sin(2t) + C \]

### Step 5: Substitute Back
Now, substitute back \( t = e^{x^3} \):
\[ = \frac{1}{6} \sin(2 e^{x^3}) + C \]

### Final Answer
The final answer is:
\[ \int x^2 e^{x^3} \cos(2 e^{x^3}) \, dx = \frac{1}{6} \sin(2 e^{x^3}) + C \]

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