Finding Integral of a Divided Function (Hyperbola?)

  • Thread starter Mathpower
  • Start date
  • #1
Mathpower
28
0

Homework Statement


Integrate between y=0 and y=-0.5:
∫((y+0.5)/(0.5-y)) dy

Homework Equations


Can you please show me how to integrate it...Then I will be able to take it from there and substitute in the appropriate values.


The Attempt at a Solution


Quotient rule in reverse?...wouldn't work
Using ln ...wouldn't work
Try to factorise...nope
Use rationalising of denominator (conjugate idea)...that just makes it even more complicated
Now I have run out of ideas ...this is out of my scope. (Scholarship paper NZ)

Thank you in advance,
Kindest Regards.
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,568
774

Homework Statement


Integrate between y=0 and y=-0.5:
∫((y+0.5)/(0.5-y)) dy

Homework Equations


Can you please show me how to integrate it...Then I will be able to take it from there and substitute in the appropriate values.


The Attempt at a Solution


Quotient rule in reverse?...wouldn't work
Using ln ...wouldn't work
Try to factorise...nope
Use rationalising of denominator (conjugate idea)...that just makes it even more complicated
Now I have run out of ideas ...this is out of my scope. (Scholarship paper NZ)

Thank you in advance,
Kindest Regards.

Hint: Divide the numerator by the denominator to make the degree of the numerator one less than the degree of the denominator.
 
  • #3
Mathpower
28
0
Hmmm...Thanks for the hint, but I don't get what you meant here...
Isn't the numerator already being divided by the denominator and aren't they to the same degree anyway?
 
  • #4
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,568
774
Write it as$$
-\frac{y + \frac 1 2}{y -\frac 1 2}$$
and do one step of a long division writing it as quotient + remainder/divisor.

Alternatively you could write it as$$
-\frac {(y-\frac 1 2)+1}{(y-\frac 1 2)}$$and break it into two terms. It's the same result either way.
 
  • #5
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,693
1,273
Hmmm...Thanks for the hint, but I don't get what you meant here...
Isn't the numerator already being divided by the denominator and aren't they to the same degree anyway?
Yes, they're the same degree, so use some sort of division algorithm to find a quotient & remainder.
 
  • #6
Mathpower
28
0
Nice! I like the alternative (because its the only thing I understood) from LCKurtz...rationalising the denominator.
The rest of it didn't make much sense...how can you integrate with a remainder! (I must be missing something)
Thank you so much for all your help.

P.S.: Sorry for not being able to understand what you guys said.

Kindest Regards
 
  • #7
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,693
1,273
Nice! I like the alternative (because its the only thing I understood) from LCKurtz...rationalising the denominator.
The rest of it didn't make much sense...how can you integrate with a remainder! (I must be missing something)
Thank you so much for all your help.

P.S.: Sorry for not being able to understand what you guys said.

Kindest Regards
Here's how that remainder works:

Write the remainder of 1 as a fraction.

[itex]\displaystyle \frac{1}{y-\frac{1}{2}}[/itex]

So that [itex]\displaystyle -\frac {(y+\frac 1 2)}{(y-\frac 1 2)}=-\left(1+\frac{1}{y-\frac{1}{2}}\right)[/itex]

Although expressions like the one here, I often use the same method as LCKurtz showed you. For more complicated situations, you should know how to express the remainder as a fraction.
 

Suggested for: Finding Integral of a Divided Function (Hyperbola?)

Replies
7
Views
394
Replies
1
Views
396
  • Last Post
Replies
3
Views
628
  • Last Post
Replies
13
Views
434
Replies
4
Views
486
Replies
4
Views
407
Replies
4
Views
747
Replies
7
Views
848
Replies
13
Views
417
Top