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.