To solve the integral of $(x^2 + 2x) \cos(x) \, dx$ using integration by parts, we need to apply the formula:

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

Step 1: Choose $u$ and $dv$

We need to select parts of the integral to assign to $u$ and $dv$. A good choice is to let:

$u = x^2 + 2x \quad \text{and} \quad dv = \cos(x) \, dx$

Step 2: Compute $du$ and $v$

Now, differentiate $u$ to get $du$ and integrate $dv$ to get $v$.

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

Step 3: Apply the integration by parts formula

Now, apply the formula:

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

Step 4: Simplify the remaining integral

We now need to integrate the term $\int \sin(x) \, (2x + 2) \, dx$. Let's separate this:

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

For the first part, we will use integration by parts again.

Step 5: Integrate $\int 2x \sin(x) \, dx$ using integration by parts

Let’s use integration by parts again for $\int 2x \sin(x) \, dx$.

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

Now, apply the integration by parts formula again:

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

The remaining integral is straightforward:

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

Thus:

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

Step 6: Combine everything

Now, combine all the results:

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

Simplifying:

$\int (x^2 + 2x) \cos(x) \, dx = (x^2 + 2x) \sin(x) + 2x \cos(x) - 4 \sin(x) + C$

Where $C$ is the constant of integration.

---

Recap:

$\int (x^2 + 2x) \cos(x) \, dx = (x^2 + 2x) \sin(x) + 2x \cos(x) - 4 \sin(x) + C$

Would you like more details or clarifications on any step?

Here are 5 related questions you might find useful:

1. What happens if we use integration by parts in a different way for this integral?
2. How do you handle higher-order polynomials in integration by parts?
3. What’s the general formula for integration by parts?
4. Can we use a tabular method for this integral? If so, how?
5. What are other methods for integrating trigonometric functions?

Tip: When integrating polynomials multiplied by trigonometric functions, always try to reduce the degree of the polynomial step by step through integration by parts.