https://file.aiquickdraw.com/mathbot/file/1729045056179k3qtdun8.jpg

The integral in the image is:

\[
\int x^2 \sin(x) \, dx
\]

To solve this, we will use **integration by parts**. The formula for integration by parts is:

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

### Step 1: Choose \( u \) and \( dv \)

We let:
- \( u = x^2 \) (so that \( du = 2x \, dx \))
- \( dv = \sin(x) \, dx \) (so that \( v = -\cos(x) \))

### Step 2: Apply the formula

Using the integration by parts formula:

\[
\int x^2 \sin(x) \, dx = u v - \int v \, du
\]

Substitute the values of \( u \), \( v \), \( du \), and \( dv \):

\[
= x^2 (-\cos(x)) - \int (-\cos(x)) (2x \, dx)
\]
\[
= -x^2 \cos(x) + 2 \int x \cos(x) \, dx
\]

### Step 3: Apply integration by parts again

For the remaining integral \( \int x \cos(x) \, dx \), we apply integration by parts again. Let:
- \( u = x \) (so that \( du = dx \))
- \( dv = \cos(x) \, dx \) (so that \( v = \sin(x) \))

Now apply the formula:

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

### Step 4: Combine the results

Now substitute back into the original equation:

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

Thus, the solution to the integral is:

\[
\boxed{-x^2 \cos(x) + 2x \sin(x) + 2\cos(x) + C}
\]

Would you like more details on any step?

Here are five related questions:
1. How would you apply integration by parts to \( \int e^x \sin(x) \, dx \)?
2. What are some common functions that require integration by parts?
3. Can you solve \( \int x^3 \cos(x) \, dx \) using integration by parts?
4. How would you derive the reduction formula for \( \int x^n e^x \, dx \)?
5. How does integration by parts work with definite integrals?

**Tip**: Integration by parts is especially useful when integrating products of algebraic and trigonometric (or exponential) functions. Always carefully choose \( u \) and \( dv \) to simplify the problem.

Learn how to solve the integral ∫x^2sin(x)dx using the integration by parts method. This detailed guide walks through step-by-step integration for advanced calculus students, including key concepts and formulas. Suitable for Grades 11-12.

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