Question

evaluate the integral: -9/(x+5)^2(x-1)dx

Answer 1

You have written \(\dfrac{-9}{(x+5)^2}\cdot (x-1)\). Is this your intent?

Answer 2

The (x-1) is also in the denominator.

-9/[(x+5)^2(x-1)]

Answer 3

Very goo correction. Go to a little extra effort to communicate clearly. It will help you every day.

Have you considered a Partial Fraction Decomposition?

Answer 4

Yes. That's the homework I'm working on, but I keep getting stuck.

Answer 5

Okay, how did you set it up? You have three factors in the denominator.

Answer 6

A/(x+5) +B/(x+5)^2+C/(x-1)

Answer 7

Perfect. So now we have A(x+5)(x-1) + B(x-1) + C(x+5)^2 = -9

What's next?

Answer 8

A(x+5)^2(x-1)+B(x+5)(x-1)+C(x+5)(x+5)^2 and then expand.

Answer 9

No. Too much stuff in there. Look at mine and see where something went wrong. We're just finding the LEAST common denominator, which we already know. No need to go farther than that. Look particularly at that last one. Why would there EVER be THREE factors of (x+5)?

Answer 10

I know there is probably an easier way to think about this, and I probably missed the overall idea behind some of the work, so I go through more steps than need be. Is there a faster way?

Answer 11

There are lots of ways to do it wrong. The expression I wrote is not that complicated. You are discouraged on this one because you have some cubic factors that don't need to be there. Just use mine and get this one done. You can relearn how to add fractions after you gain a little confidence.

Answer 12

So, just expand the equation you wrote above?

Answer 13

Yes. There is nothing worse than a quadratic in there.

Answer 14

Note: These CAN be a little tedious. Hang in there and be careful.

Answer 15

So I have three equations now:

x^2(A+C) , x(4A+B+10C) , (-5A-B+25C)

Answer 16

Excellent. Good job wading through it.

For polynomials to be identical ALL their coefficients must match. We're matching with just "-9", so all coefficients on powers of x need to be zero.

A + C = 0
4A + B + 10C = 0
-5A - B + 25C = -9

Solve the system!

Answer 17

Yes! That's where I kept messing up. I couldn't figure out how each equations could match with the same -9. Thank you, thank you.

Answer 18

Well, there you have it, then.