https://file.aiquickdraw.com/mathbot/file/1729028852657lp4h7wxs.jpg

The image contains a mathematical problem involving the integral:

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

This is a typical integration problem that requires the use of **integration by parts**. Here's how to solve it:

### Step-by-Step Solution (Integration by Parts):

We use the formula for integration by parts:

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

1. **Choose** \( u = x \) and \( dv = \sin(x) \, dx \).

2. **Differentiate** \( u \) and integrate \( dv \):
   - \( du = dx \)
   - \( v = -\cos(x) \)

3. **Substitute** into the integration by parts formula:

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

4. Now, integrate \( \int \cos(x) \, dx \):

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

5. Combine the results:

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

### Final Answer:

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

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How is integration by parts derived from the product rule of differentiation?
2. Can integration by parts be applied more than once in a problem?
3. How would you approach integrating \( \int x^2 \sin(x) \, dx \)?
4. What are the key differences between integration by parts and substitution methods?
5. How can integration by parts be used in definite integrals?

**Tip**: Always double-check which function to choose as \( u \) and \( dv \) for integration by parts. Functions that simplify upon differentiation, like \( x \), often make good choices for \( u \).

Learn how to solve the integral of x*sin(x) dx using the integration by parts technique. This problem demonstrates key calculus concepts, such as product rule and integration strategies, suitable for Grade 12 or early college students.

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