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The integral to evaluate is:

\[
\int_{2}^{4} \frac{6x^2 - 10x^4}{x^5 - x^3} \, dx
\]

### Step 1: Determine whether the integral is improper

An integral is considered improper if:

1. The limits of integration include points where the integrand becomes infinite (such as vertical asymptotes).
2. One or both of the limits of integration approach infinity.

Let’s examine the integrand:

\[
f(x) = \frac{6x^2 - 10x^4}{x^5 - x^3} = \frac{6x^2 - 10x^4}{x^3(x^2 - 1)} = \frac{6x^2 - 10x^4}{x^3(x-1)(x+1)}
\]

We need to determine whether there are any points where the denominator equals zero in the interval \([2, 4]\).

The denominator, \( x^3(x-1)(x+1) \), has potential zeros at \(x = 0\), \(x = 1\), and \(x = -1\).

- \(x = 0\) is outside the interval \([2, 4]\).
- \(x = 1\) is also outside the interval.
- \(x = -1\) is outside the interval.

Since the integrand does not become infinite within the interval \([2, 4]\) and the limits are finite, **the integral is not improper**.

### Step 2: Simplifying the Integrand

We can simplify the integrand further:

\[
\frac{6x^2 - 10x^4}{x^3(x-1)(x+1)} = \frac{6 - 10x^2}{(x-1)(x+1)}
\]

Now, the integrand is much simpler:

\[
f(x) = \frac{6 - 10x^2}{x(x-1)(x+1)}
\]

### Step 3: Evaluate the Integral

Now we will attempt to evaluate the integral using partial fractions.

Let me compute this.The simplified integrand is:

\[
f(x) = \frac{-2(5x^2 - 3)}{x(x-1)(x+1)}
\]

The evaluated integral from 2 to 4 is:

\[
\int_2^4 \frac{-2(5x^2 - 3)}{x(x-1)(x+1)} \, dx = -6 \ln(4) - 2 \ln(15) + 2 \ln(3) + 6 \ln(2)
\]

This expression can be simplified further using properties of logarithms, but it is the exact result of the integral.

Do you want the final simplification or any further details?

Here are 5 related questions:
1. What makes an integral improper?
2. How do you approach integrals with rational functions like this one?
3. Why is factoring the denominator useful in evaluating integrals?
4. How do partial fractions work in integration?
5. Can you provide a real-life example where improper integrals are used?

**Tip:** Logarithmic properties can help simplify complex logarithmic expressions during integration.

Determine whether the following integrals are improper or not and state the reason of your answer. Then evaluate the following integral: ∫(6x^2 - 10x^4)/(x^5 - x^3) dx, from x=2 to x=4.

This integral involves the evaluation of a rational function over a finite interval. Learn how to determine if an integral is improper and how to apply partial fractions for solution. Suitable for calculus students.

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