evaluate  the integral  in the limits of  9 and 4   5x squared -8x+1/2x(x-1)squared

To evaluate the given integral of the expression:

\[
\int_4^9 \frac{5x^2 - 8x + 1}{2x(x - 1)^2} \, dx
\]

We will follow these steps:

### Step 1: Simplify the Expression
First, simplify the integrand if possible, breaking the expression into more manageable parts. Let's rewrite the given expression:

\[
\frac{5x^2 - 8x + 1}{2x(x - 1)^2}
\]

### Step 2: Substitution or Decomposition
Depending on the structure of the expression, we may use techniques like substitution or partial fractions decomposition to simplify the integration. 

Let me go ahead and work this out for you!The evaluated integral is:

\[
\int_4^9 \frac{5x^2 - 8x + 1}{2x(x - 1)^2} \, dx = -2\log(3) - \frac{1}{2}\log(4) - \frac{5}{24} + \frac{1}{2}\log(9) + 2\log(8)
\]

This is the exact result, involving logarithmic terms. You can simplify this expression further if needed by approximating the logarithmic values.

Would you like to simplify this further or have any specific questions? 

Here are 5 related questions for practice:
1. How do you simplify logarithmic expressions?
2. What is the derivative of \(\frac{5x^2 - 8x + 1}{2x(x - 1)^2}\)?
3. How do you perform partial fraction decomposition?
4. What are the applications of definite integrals?
5. How can substitution help in solving integrals?

**Tip:** Breaking down complex rational functions often involves partial fractions to make integration easier.

evaluate the integral in the limits of 9 and 4  5x squared -8x+1/2x(x-1)squared

This problem involves evaluating the definite integral of the expression (5x^2 - 8x + 1)/(2x(x-1)^2) from 4 to 9 using integral calculus techniques. Learn how to apply partial fraction decomposition and logarithmic properties in this advanced calculus example, suitable for undergraduate or advanced high school students.

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