The image you provided shows a mathematical expression involving the integral of the function $f(x)$. The expression appears as follows:

$\int x^2 \sin(x) \, dx$

This is an integral that can be solved using integration by parts due to the product of $x^2$ and $\sin(x)$.

Solution Outline

Using integration by parts, we can let:

• $u = x^2$ and $dv = \sin(x) \, dx$.
• Then $du = 2x \, dx$ and $v = -\cos(x)$.

The integration by parts formula is: $\int u \, dv = uv - \int v \, du$

Applying this approach and repeating integration by parts as necessary, we can work through the solution.

Would you like the detailed step-by-step solution for this integral, or do you have any specific questions about integration by parts?

Here are some related questions that might deepen your understanding:

1. What are the steps to apply integration by parts to integrals with powers of $x$ and trigonometric functions?
2. How does the process change if we have $x^3 \sin(x)$ instead of $x^2 \sin(x)$?
3. Can integration by parts be used for integrals involving products of exponential and trigonometric functions?
4. How do we determine the choice of $u$ and $dv$ in integration by parts?
5. What are alternative methods for solving integrals involving trigonometric functions and powers of $x$?

Tip: When dealing with integration by parts, choosing $u$ and $dv$ correctly (often using LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) can simplify the process.