Evaluate the integral below using integration by parts given that u=ln(5x2) and dv=2x5dx.

∫2x5ln(5x2)dx

Be sure to place the argument of the logarithmic function in parentheses in your answer.

To evaluate the integral \( \int 2x^5 \ln(5x^2) \, dx \) using integration by parts, we are given:
\[ u = \ln(5x^2) \]
\[ dv = 2x^5 \, dx \]

We need to find \( du \) and \( v \). 

First, let's find \( du \):
\[ u = \ln(5x^2) \]
To differentiate \( u \), we use the chain rule:
\[ \ln(5x^2) = \ln(5) + \ln(x^2) = \ln(5) + 2\ln(x) \]
\[ \frac{d}{dx} \ln(5x^2) = \frac{d}{dx} (\ln(5) + 2\ln(x)) \]
\[ du = 2 \cdot \frac{1}{x} \, dx = \frac{2}{x} \, dx \]

Now, let's find \( v \):
\[ dv = 2x^5 \, dx \]
Integrating \( dv \) with respect to \( x \):
\[ v = \int 2x^5 \, dx = \frac{2}{6} x^6 = \frac{1}{3} x^6 \]

Now we have:
\[ u = \ln(5x^2) \]
\[ du = \frac{2}{x} \, dx \]
\[ dv = 2x^5 \, dx \]
\[ v = \frac{1}{3} x^6 \]

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get:
\[ \int 2x^5 \ln(5x^2) \, dx = \ln(5x^2) \cdot \frac{1}{3} x^6 - \int \left( \frac{1}{3} x^6 \cdot \frac{2}{x} \right) dx \]
\[ = \frac{1}{3} x^6 \ln(5x^2) - \int \frac{2}{3} x^5 \, dx \]
\[ = \frac{1}{3} x^6 \ln(5x^2) - \frac{2}{3} \int x^5 \, dx \]
\[ = \frac{1}{3} x^6 \ln(5x^2) - \frac{2}{3} \cdot \frac{1}{6} x^6 \]
\[ = \frac{1}{3} x^6 \ln(5x^2) - \frac{1}{9} x^6 + C \]

Therefore, the final answer is:
\[ \int 2x^5 \ln(5x^2) \, dx = \frac{1}{3} x^6 \ln(5x^2) - \frac{1}{9} x^6 + C \]

Would you like more details or have any questions?

Here are five more questions you might want to ask:
1. How do you integrate \( \int x e^x \, dx \) using integration by parts?
2. Can you explain how to solve a second-order differential equation?
3. What is the method of substitution in integration?
4. How do you find the Maclaurin series of a function?
5. What are the properties of improper integrals?

**Tip:** When dealing with logarithmic functions in integration by parts, make sure to carefully handle the argument of the logarithmic function to avoid mistakes in differentiation.

Learn how to evaluate the integral ∫2x^5ln(5x^2)dx using integration by parts. This problem involves logarithmic functions and requires application of the integration by parts formula. Suitable for college-level mathematics.

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