Integration by partial Fractions question

[bns1201]

Homework Statement

The problem asks to evaluate the integral using partial fractions, but I just cannot find out which trick to get this one to work. the equation is
$$\int$$$$\frac{x^3+x^2+2x+1}{(x^2+1)(x^2+2)}$$

The Attempt at a Solution

I have tried setting it up as a partial fraction with each part of the numerator under Ax+B and Cx+D but I have a feeling that is incorrect. Thanks in advance!

 

[Dick]

Don't let a feeling it's incorrect stop you from doing it. If you followed through with your original program, you'd be done now. That's perfectly correct.

 

[bns1201]

When I match up the coefficients to the problem i then get

A+C=1

B+D=1

2A+C=2

2B+D=1


This is where I get stuck again because how can B+D and 2B+D both equal 1?

 

[Dick]

They can both be equal to 1 if B=0 and D=1. Don't overthink it. Just solve the equations. Subtract B+D=1 from 2B+D=1 and you'll reach that conclusion.

 

[bns1201]

thanks...i always over think the problems and forget about 0...thanks so much. calc 2 is really kickin my butt

 

[Dick]

You could kick it's butt instead. Everything you posted was exactly correct. Just push ahead and try it before you start thinking you must be wrong.

 

[bns1201]

I'm having problems with another integral of similar properties.

The integral is
$$\int$$$$\frac{1}{x\sqrt{x+1}}$$

I made this into a u sub, where u = $$\sqrt{x+1}$$ and u^2 = x+1 Therefore x+u^2-1


So the integral is now $$\int$$$$\frac{1}{u^3-u}$$ which i made into $$\int$$$$\frac{1}{u(u-1)(u+1)}$$

I then used partial fractions to make it $$\frac{A}{u}$$ + $$\frac{B}{u+1}$$+$$\frac{C}{u-1}$$


Getting me A=-1 B=$$\frac{1}{2}$$ and C=$$\frac{1}{2}$$

Which changed to equation into $$\int$$$$\frac{-1}{u}$$+$$\frac{1}{2}$$$$\int$$$$\frac{du}{u+1}$$+$$\frac{1}{2}$$$$\int$$$$\frac{du}{u-1}$$


Which gets me to the answer -ln|$$\sqrt{x+1}$$|+ 1/2 ln|$$\sqrt{x+1}+1$$|+1/2 ln |$$\sqrt{x+1}-1$$|

but the answer is ln [tex]\frac{$$\sqrt{x+1}+1}{\sqrt{x+1}-1}$$<br /> <br /> <br /> <br /> <br /> <br /> Can anyone provide me with help as to where I am messing up? Thanks[/tex]

 

[Mark44]

In your original integral you omitted dx. In your substitution, you neglected to calculate du, which is not equal to dx.

 

[bns1201]

Thanks, I'll redo the equation right now
EDIT: ahhh I did it over with dx in there...thanks a bunch

 

[bns1201]

Another question please:
I have to do

$$\int$$$$\frac{2x}{(x-3)^2}$$ with upper bound 2 and lower bound 0

What I've done so far:

Partial Fractions, making it: $$\frac{A}{x-3}$$+$$\frac{B}{(x-3)^2}$$

=> A(x-3)+B=2x
=> A=2 B=6
=> $$\int$$$$\frac{2}{x-3}$$ + $$\frac{6}{((x-3)^2)}$$
=> 2 ln|x-3| - 6 ln|t-3| again, evaluated from 0 to 2

I end up with -2 ln|3| + 6 ln|3|

The answer is 4-ln|9| Can anyone tell me where I'm going wrong? Thanks

 

[Mark44]

Your partial fraction decomposition was fine, but your 2nd integral wasn't.
$$\int \frac{6}{(x - 3)^2}dx \neq ln|(x - 3)^2| + C$$

Also, you could have done the problem more simply by using an ordinary substitution: u = x - 3, du = dx. You would have gotten u + some stuff in the numerator and u^2 in the denominator, which can be integrated easily.

 

[bns1201]

Thanks with the hint on u substitution, that was a lot easier


Another question if possible.

How would you start the integral

$$\int$$$$\frac{e^arctany)}{1+y^2}$$

with boundries of -1 to 1

I can't seem to find the trick...

 

[Mark44]

u = arctan(y).
What is du?

If you omit du, you're hosed.

Hint: pay attention to and don't lose those differentials!

 

[bns1201]

ahhh yes i see it now. Thanks a bunch.

 

[bns1201]

Can anyone help me as to where to start this one out? Thanks


$$\int$$(1+$$\sqrt{x}$$)^8 dx

 

[Dick]

Start with the obvious and unavoidable u=1+sqrt(x). Where does that take you?

 

[bns1201]

thanks again I appreciate it