Question

I'm encountering a sign issue in what I think is a pretty simple problem... Would someone be willing to take a look, and show me what I'm doing wrong?

The problem: "Evaluate: (integral)[(v^2 + 3v^6)/v^4]dv, from 1 to 2."

Any and all help is greatly appreciated!

Answer 1

Here's what I did:
\[f(v) = \frac{v^2 + 3v^6}{v^4}\] = \[f(v) = \frac{v^2 + 3v^6}{v^4} = \frac{3v^4 + 1}{v^2}\]
\[= (3v^4 + 1)v^{-2}\]
\[= 3v^2 + v^{-2}\]
\[F(v) = v^3 + (-v^{-1})\]
\[= v^3 - \frac{1}{v} \]

Answer 2

looks good to me

Answer 3

Now here is where I genuinely kind of kind don't understand what my book does. The final answer is: ln(2) + 7, meaning that F(b) must have a ln(2) in it. The obvious thing in that the v^(-1) is substituted by a ln(v), but wouldn't the v^(-1) have to be in f(v), not a term in F(v), for that to happen?
The DERIVATIVE of ln(v) = 1/v, but they're not equal in and of themselves, so why is a ln(v) showing up in the evaluation if there isn't a 1/v in the function –> f(v) –> 3v^2 + v^(-2) ??

Answer 4

Well, I took the book at its work, and went ahead plugging in a "ln(v)" for v^(-1), and got the following at the end, after doing this...

Answer 5

there is no log in the answer

Answer 6

maybe it is a typo in the question, or you are looking at the wrong answer
the only way for there to be a log is if you had \(\frac{1}{v}\) in the integrand

Answer 7

\[= F(b) - F(a)\]
\[[(2^3) - \ln(2)] - [(1^3)-\ln(1)]\]
\[(8-\ln(2)) - (1)\]
\[7 - \ln(2)\]

Answer 8

nope

Answer 9

http://www.wolframalpha.com/input/?i=integral+1+to+2++ \frac{v^2+%2B+3v^6}{v^4}

Answer 10

don't think any more about it
it is wrong that is all

Answer 11

I totally agree @satellite73 !!
I've double, and triple checked that I've written down the right problem and am looking at the right answer, and even checked the answer on wolframalpha (they say 15/2 which is darn close to [ln(2) + 7] in a calculator.
I don't know what I did wrong, but my book has 7 PLUS ln(2), and I got 7 MINUS ln(2) .... :/

Answer 12

it happens and people get bend out of shape because they believe the book as to be right so they must have screwed up somewheres
think no more about it the answer is 7.5

Answer 13

the question was not written by god or even a mathematician
neither were the answers
they are written by impoverished graduate students exhausted from a diet of ramen noodles and discount beer

Answer 14

When I say screw the book's "ln(v)" I get 7.5 exactly as well. Thank you so much!!!

Answer 15

Bahahaha!!! I love that reality check–– made my night;)

Answer 16

now forget about it

Answer 17

I had the ramen noodles ... I skipped the discount beer

Answer 18

add soy sauce, hot sauce to splurge

Answer 19

to the noodles and the beer

Answer 20

You guys are the best;) Thanks again!

Answer 21

ramen noodles and discount beer, LOL

Answer 22

I'm more of a ricaroni and cheep vodka kind of guy...

Answer 23

Glad to know not all the mods are robots, love the sarcasm ;p