To find the integral \(\int 2x^2 \cos(4x) \, dx\), we can use the method of integration by parts. This method states:

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

For this problem, we can set \( u = 2x^2 \) and \( dv = \cos(4x) \, dx \).

First, we need to compute \( du \) and \( v \):
\[
u = 2x^2 \quad \Rightarrow \quad du = 4x \, dx
\]
\[
dv = \cos(4x) \, dx \quad \Rightarrow \quad v = \frac{\sin(4x)}{4}
\]

Now we can apply the integration by parts formula:
\[
\int 2x^2 \cos(4x) \, dx = 2x^2 \cdot \frac{\sin(4x)}{4} - \int \frac{\sin(4x)}{4} \cdot 4x \, dx
\]
\[
= \frac{2x^2 \sin(4x)}{4} - \int x \sin(4x) \, dx
\]
\[
= \frac{x^2 \sin(4x)}{2} - \int x \sin(4x) \, dx
\]

We now need to integrate \(\int x \sin(4x) \, dx\) using integration by parts again. Let \( u = x \) and \( dv = \sin(4x) \, dx \):
\[
u = x \quad \Rightarrow \quad du = dx
\]
\[
dv = \sin(4x) \, dx \quad \Rightarrow \quad v = -\frac{\cos(4x)}{4}
\]

Applying the integration by parts formula again:
\[
\int x \sin(4x) \, dx = x \left( -\frac{\cos(4x)}{4} \right) - \int \left( -\frac{\cos(4x)}{4} \right) \, dx
\]
\[
= -\frac{x \cos(4x)}{4} + \frac{1}{4} \int \cos(4x) \, dx
\]
\[
= -\frac{x \cos(4x)}{4} + \frac{1}{4} \cdot \frac{\sin(4x)}{4}
\]
\[
= -\frac{x \cos(4x)}{4} + \frac{\sin(4x)}{16}
\]

Combining both parts, we have:
\[
\int 2x^2 \cos(4x) \, dx = \frac{x^2 \sin(4x)}{2} - \left( -\frac{x \cos(4x)}{4} + \frac{\sin(4x)}{16} \right)
\]
\[
= \frac{x^2 \sin(4x)}{2} + \frac{x \cos(4x)}{4} - \frac{\sin(4x)}{16} + C
\]

Thus, the integral \(\int 2x^2 \cos(4x) \, dx\) is:
\[
\int 2x^2 \cos(4x) \, dx = \frac{x^2 \sin(4x)}{2} + \frac{x \cos(4x)}{4} - \frac{\sin(4x)}{16} + C
\]

where \( C \) is the constant of integration.

Would you like more details or have any questions about this solution?

Here are 5 numbered questions you might ask:
1. How does the method of integration by parts work?
2. How do you choose \( u \) and \( dv \) in integration by parts?
3. Can you explain the steps to integrate \( \cos(4x) \) and \( \sin(4x) \)?
4. What is the general strategy for integrating products of polynomials and trigonometric functions?
5. How do you verify the result of an integral?

**Tip:** For complex integrals involving products of polynomial and trigonometric functions, consider using integration by parts multiple times as needed.

Learn how to integrate 2x^2 cos(4x) using integration by parts. This detailed step-by-step solution covers trigonometric and polynomial integrals, suitable for advanced college-level mathematics.

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