The integral in the image is:

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

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

$\int u \, dv = uv - \int v \, du$

We need to choose $u$ and $dv$ wisely. Let:

• $u = x^2$ so that $du = 2x \, dx$
• $dv = \cos(x) \, dx$, which gives $v = \sin(x)$

Now applying the formula:

$\int x^2 \cos(x) \, dx = x^2 \sin(x) - \int 2x \sin(x) \, dx$

We now need to compute $\int 2x \sin(x) \, dx$ using integration by parts again.

Let:

• $u = 2x$, so $du = 2 \, dx$
• $dv = \sin(x) \, dx$, so $v = -\cos(x)$

Applying the formula again:

$\int 2x \sin(x) \, dx = -2x \cos(x) + \int 2 \cos(x) \, dx$

Now compute the remaining integral:

$\int 2 \cos(x) \, dx = 2 \sin(x)$

Thus, the integral becomes:

$\int x^2 \cos(x) \, dx = x^2 \sin(x) - \left( -2x \cos(x) + 2 \sin(x) \right)$

Simplifying:

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

This is the solution to the integral.

Would you like further clarification or details on any step?

Here are five related questions for further exploration:

1. How does integration by parts work, and why is it useful for certain integrals?
2. Can the order of choosing $u$ and $dv$ in integration by parts affect the outcome?
3. What are some common integrals involving trigonometric functions and polynomials?
4. How can you verify the result of an integral?
5. What other techniques besides integration by parts could be used for integrals involving trigonometric functions?

Tip: When choosing $u$ in integration by parts, it's often beneficial to pick the function that simplifies upon differentiation.