If the integral of 9x-2 /(x-3)(2x-1)dx= (A)Ln|x – 3| + (B)Ln|2x – 1| + constant, then what is the value of A – B?

a. 3
b. 4.5
c. 5.5
d. 4

I get 5.5

What do you get?

I don't see a choice for you answer to this "partial fractions" question.

To check if you have the correct values of A and B,
evaluate a(2x-1) + B(x-3)
you should get 9x - 2
Then simply find A-B

btw, I did not get any of the given choices.

Let me know what you found for A and B

A(2x-1) + B(x-3) = 9x - 2

let x = 3, 5A + 0 = 25 ----> A = 5
let x = 1/2
0 - (5/2)B = 5/2 -----> B = -1
A - B = 5 - (-1) = 6

To find the value of A - B, we need to compare the given integral with the given expression (A)Ln|x – 3| + (B)Ln|2x – 1| + constant.

Let's analyze the integral step by step to determine the value of A and B.

First, we need to find the two antiderivatives (integrals) of the given expression: 9x-2 /(x-3)(2x-1)dx.

To do this, we can use partial fraction decomposition.

1. Write the expression as a sum of two fractions:

9x - 2 = A / (x - 3) + B / (2x - 1)

2. Multiply both sides by (x - 3)(2x - 1) to eliminate the denominators:

(9x - 2)(x - 3)(2x - 1) = A(2x - 1) + B(x - 3)

Expanding and simplifying this equation gives:

18x^3 - 47x^2 + 36x - 2 = (2A + B)x - (A + 3B)

By comparing the coefficients of the x terms on both sides of the equation, we get:

2A + B = 18 (1)
A + 3B = -2 (2)

Solving this system of equations will give us the values of A and B.

Multiplying equation (2) by 2, we get:

2A + 6B = -4 (3)

Now we can subtract equation (1) from equation (3):

2A + 6B - (2A + B) = -4 - 18

Simplifying,

5B = -22

Dividing both sides by 5,

B = -22 / 5

Now substituting the value of B in equation (2), we get:

A + 3(-22/5) = -2

A - (66/5) = -2

A = -2 + 66/5

A = (10 - 66/5)/5

A = (50/5 - 66/5)/5

A = -16/5

Therefore, we have A = -16/5 and B = -22/5.

Finally, we can find the value of A - B:

A - B = -16/5 - (-22/5)

A - B = -16/5 + 22/5

A - B = 6/5

So, the value of A - B is 6/5.

This is not one of the options given, so none of the options a, b, c, or d is the correct answer.