To integrate \( x^2 \sin(x) \), we will use **integration by parts**, which is based on the formula:

\[
\int u \, dv = uv - \int v \, du
\]

### Step 1: Set up the parts
We need to choose \( u \) and \( dv \) such that:
- \( u = x^2 \) (since it simplifies when differentiated)
- \( dv = \sin(x) \, dx \) (since the integral of \( \sin(x) \) is easy to compute)

### Step 2: Differentiate and Integrate
Now, differentiate \( u \) and integrate \( dv \):
- \( du = 2x \, dx \)
- \( v = -\cos(x) \) (since \( \int \sin(x) \, dx = -\cos(x) \))

### Step 3: Apply the integration by parts formula
\[
\int x^2 \sin(x) \, dx = -x^2 \cos(x) - \int -2x \cos(x) \, dx
\]
\[
= -x^2 \cos(x) + 2 \int x \cos(x) \, dx
\]

### Step 4: Apply integration by parts again on \( \int x \cos(x) \, dx \)
We need to apply integration by parts to \( \int x \cos(x) \, dx \):
- Let \( u = x \) and \( dv = \cos(x) \, dx \)
- Then, \( du = dx \) and \( v = \sin(x) \)

So,
\[
\int x \cos(x) \, dx = x \sin(x) - \int \sin(x) \, dx
\]
\[
= x \sin(x) + \cos(x)
\]

### Step 5: Substitute back
Now substitute this back into the original integral:
\[
\int x^2 \sin(x) \, dx = -x^2 \cos(x) + 2 \left( x \sin(x) + \cos(x) \right)
\]
\[
= -x^2 \cos(x) + 2x \sin(x) + 2 \cos(x) + C
\]

### Final Answer:
\[
\int x^2 \sin(x) \, dx = -x^2 \cos(x) + 2x \sin(x) + 2 \cos(x) + C
\]

Would you like more details or have any questions? Here's a tip:

### Tip:
When performing integration by parts, always try to pick \( u \) to be the part that simplifies when differentiated, and \( dv \) to be the part whose integral is easy to compute.

### Related Questions:
1. What if we need to integrate \( x^3 \sin(x) \)? 
2. How do we handle an integral with more complex functions like \( e^x \sin(x) \)?
3. Can integration by parts be used to solve definite integrals as well?
4. What happens if the function is \( \cos(x) \) instead of \( \sin(x) \)?
5. What if we use the reduction formula for trigonometric integrals?

Learn how to solve the integral of x^2sin(x) using the integration by parts method. This detailed step-by-step guide covers key concepts in integration and trigonometric functions, including applying integration by parts twice. Suitable for students in Grades 11-12.

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