The integral for image one was 5x^4ln(2x)

Thank you for providing the correct integral \(\int 5x^4 \ln(2x) \, dx\). We will use the integration by parts method to solve this integral. 

Using the integration by parts formula:
\[
\int u \, dv = uv - \int v \, du
\]

Let's choose:
\[
u = \ln(2x) \quad \text{and} \quad dv = 5x^4 \, dx
\]

First, we need to find \(du\) and \(v\):

\[
du = \frac{d}{dx} \ln(2x) \, dx = \frac{1}{2x} \cdot 2 \, dx = \frac{1}{x} \, dx
\]

\[
v = \int 5x^4 \, dx = 5 \cdot \frac{x^5}{5} = x^5
\]

Now, apply the integration by parts formula:
\[
\int 5x^4 \ln(2x) \, dx = \ln(2x) \cdot x^5 - \int x^5 \cdot \frac{1}{x} \, dx
\]

Simplify the integral:
\[
= x^5 \ln(2x) - \int x^4 \, dx
\]

\[
= x^5 \ln(2x) - \frac{x^5}{5} + C
\]

Thus, the solution to the integral \(\int 5x^4 \ln(2x) \, dx\) is:
\[
x^5 \ln(2x) - \frac{x^5}{5} + C
\]

Would you like more details or have any questions about this solution? Here are some questions you might consider:

1. Can you show how to differentiate \(\ln(2x)\) step by step?
2. How can we verify the solution by differentiating the result?
3. What are some other examples of integrals solved using integration by parts?
4. How does the choice of \(u\) and \(dv\) affect the complexity of the integral?
5. Can we use substitution methods along with integration by parts for more complex integrals?

Tip: When choosing \(u\) and \(dv\) in integration by parts, always select \(u\) such that its derivative \(du\) simplifies the integral.

Learn how to solve the integral ∫ 5x^4 ln(2x) dx using the integration by parts method. This problem involves the integration by parts formula and the differentiation of natural logarithm. Suitable for undergraduate level mathematics.

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