`intx^(2) . sin x^(3) dx`

Answer

To solve the integral \( \int x^2 \sin(x^3) \, dx \), we will use the method of substitution. Here is the step-by-step solution:

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

### Step 2: Substitute in the integral
Now, we can express \( x^2 \) in terms of \( t \):
\[ x^2 = \left( \frac{t}{x} \right)^{\frac{2}{3}} \quad \text{(but we will keep it as is for now)} \]
Substituting \( t \) and \( dx \) into the integral:
\[ \int x^2 \sin(x^3) \, dx = \int x^2 \sin(t) \cdot \frac{dt}{3x^2} \]
This simplifies to:
\[ \int \sin(t) \cdot \frac{dt}{3} = \frac{1}{3} \int \sin(t) \, dt \]

### Step 3: Integrate
The integral of \( \sin(t) \) is:
\[ \int \sin(t) \, dt = -\cos(t) + C \]
Thus,
\[ \frac{1}{3} \int \sin(t) \, dt = -\frac{1}{3} \cos(t) + C \]

### Step 4: Substitute back
Now, we substitute back \( t = x^3 \):
\[ -\frac{1}{3} \cos(t) + C = -\frac{1}{3} \cos(x^3) + C \]

### Final Answer
Therefore, the integral \( \int x^2 \sin(x^3) \, dx \) evaluates to:
\[ -\frac{1}{3} \cos(x^3) + C \]

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