# Finding Integral of a Divided Function (Hyperbola?)

1. Sep 21, 2012

### Mathpower

1. The problem statement, all variables and given/known data
Integrate between y=0 and y=-0.5:
∫((y+0.5)/(0.5-y)) dy

2. Relevant 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.

3. 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)

Kindest Regards.

2. Sep 21, 2012

### LCKurtz

Hint: Divide the numerator by the denominator to make the degree of the numerator one less than the degree of the denominator.

3. Sep 21, 2012

### Mathpower

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. Sep 21, 2012

### LCKurtz

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. Sep 21, 2012

### SammyS

Staff Emeritus
Yes, they're the same degree, so use some sort of division algorithm to find a quotient & remainder.

6. Sep 21, 2012

### Mathpower

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. Sep 21, 2012

### SammyS

Staff Emeritus
Here's how that remainder works:

Write the remainder of 1 as a fraction.

$\displaystyle \frac{1}{y-\frac{1}{2}}$

So that $\displaystyle -\frac {(y+\frac 1 2)}{(y-\frac 1 2)}=-\left(1+\frac{1}{y-\frac{1}{2}}\right)$

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.