To compute the indeﬁnite integral

∫(6x^4+9/x^2+1)dx

a) Perform the long division to get

6x^4+9/x^2+1=A(x)+B(x)/x^2+1

where  A(x)
and  B(x)
are two polynomials in  x
or constants.

Answer:

A(x)=?

B(x)=?

We are tasked with finding \( A(x) \) and \( B(x) \) by performing polynomial long division for the expression:

\[
6x^4 + \frac{9}{x^2} + 1
\]

Let's first rewrite this expression in a form suitable for long division:

\[
6x^4 + \frac{9}{x^2} + 1 = 6x^4 + 1 + \frac{9}{x^2}
\]

The goal is to express this as \( A(x) + \frac{B(x)}{x^2} \), where \( A(x) \) is a polynomial and \( B(x) \) is another polynomial or constant. We start by dividing the polynomial part \( 6x^4 + 1 \) by \( x^2 \), ignoring the fraction term for now.

### Step 1: Perform Long Division of \( 6x^4 + 1 \) by \( x^2 \)

1. Divide the first term of the dividend, \( 6x^4 \), by \( x^2 \). This gives \( 6x^2 \).
2. Multiply \( 6x^2 \) by \( x^2 \), which gives \( 6x^4 \).
3. Subtract \( 6x^4 \) from \( 6x^4 + 1 \), leaving a remainder of \( 1 \).

Thus, we have:

\[
6x^4 + 1 = 6x^2 + \frac{1}{x^2}
\]

Now, combining everything, we can rewrite the original expression as:

\[
6x^4 + \frac{9}{x^2} + 1 = 6x^2 + \frac{1 + 9}{x^2} = 6x^2 + \frac{10}{x^2}
\]

### Step 2: Identify \( A(x) \) and \( B(x) \)

From this result, we can see that:

- \( A(x) = 6x^2 \) (the polynomial part)
- \( B(x) = 10 \) (the constant in the numerator of the fractional part)

Thus, the final answer is:

\[
A(x) = 6x^2
\]
\[
B(x) = 10
\]

---

Do you want further details or explanations? Let me know if you have any questions!

Here are 5 related questions:
1. How do you perform polynomial long division step-by-step?
2. What is the next step in finding the indefinite integral of this function?
3. Can we simplify the integral further before solving?
4. What techniques are useful for integrating rational functions?
5. How does polynomial long division help in integration?

**Tip:** When dealing with rational functions, try polynomial long division if the degree of the numerator is higher than the degree of the denominator—it simplifies the integral.

To compute the indefinite integral ∫(6x^4 + 9/x^2 + 1) dx. a) Perform the long division to get 6x^4 + 9/x^2 + 1 = A(x) + B(x)/x^2 + 1, where A(x) and B(x) are two polynomials in x or constants. Answer: A(x)=? B(x)=?

Learn how to compute the indefinite integral of the expression (6x^4 + 9/x^2 + 1) by performing polynomial long division. This problem covers advanced calculus concepts like polynomial long division and integration of rational functions. Ideal for students in Grades 11-12.

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