`intx^(2)sinx^(3)dx=?`

Options

A. `cosx^(3)+C`
B. `-cosx^(3)+C`
C. `-(1)/(3)cosx^(3)+C`
D. none of these

Explanation

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

### Step 1: Identify the substitution
Let \( t = x^3 \). Then, we need to find \( dx \) in terms of \( dt \).

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

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

### Step 4: Simplify the integral
Notice that \( x^2 \) in the numerator and denominator cancels out:
\[ = \int \sin(t) \cdot \frac{dt}{3} = \frac{1}{3} \int \sin(t) \, dt \]

### Step 5: Integrate
The integral of \( \sin(t) \) is:
\[ \int \sin(t) \, dt = -\cos(t) \]
Thus,
\[ \frac{1}{3} \int \sin(t) \, dt = \frac{1}{3} (-\cos(t)) = -\frac{1}{3} \cos(t) \]

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

### Step 7: Add the constant of integration
Finally, we add the constant of integration \( C \):
\[ \int x^2 \sin(x^3) \, dx = -\frac{1}{3} \cos(x^3) + C \]

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

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