Reviewing an integral problem

[xia]

Greetings to everyone,

I am currently a HS senior following a program where I have to study independently at home and attend school for tests and exams, and I also have a limited opportunity to interact with the teachers. My Maths teacher, specifically, is very reluctant to respond to my questions via e-mail, as well as I do not have anyone else to ask for guidance regarding some calculus problems. Of course, I do not mean to offend my Maths teacher, it is just the way it is and I must find a way to perfect my knowledge without it.

Some of the problems surpass HS level Maths and are not required for examinations, so I am unable to find guidance through the books I have (if you have any recommendations for specific texts on this, I would be very interested and grateful, as I am interested in extra content).

I have attached the specific problem to this thread. I'd be very grateful if someone could review my process here and point out any mistakes I am making in my reasoning. I would also like to ask if it would be OK for me to post these types of questions occasionally, considering my situation, so I could learn more? I am not sure if there is a specific limit to such posts.

Thank you very much in advance.

[xia]

The right answer, according to Wolfram Alpha, is -(x^2 (-ln(x^2-4))+4 ln(x^2-4)+4)/(x^2-4)+constant
Alternate form: -4/( x^2 - 4 ) + ln[ x^2 - 4]

Simplifying 8 - 4x + (x^4)/2 = (x^2-4)^2
If ln[ x^2 - 4]^2 = 2*ln[ x^2 - 4] and 0.5*2*ln[ x^2 - 4] = ln[ x^2 - 4]
Which seems to imply that the first part of my answer is correct, as well as the reasoning on -4/( x^2 - 4 ), where I worried that my approach of integrating with two variables could be wrong, as I expressed dx through the differential of a new variable, thus making the other variable, x, a constant - however, according to Wolfram Alpha, it should be correct.
Then it seems that my mistakes are:
1) Multiplying both integrals by 0.5, however, I don't understand how this could be a mistake in this instance, as the fraction 0.5 should apply to both - which means that my whole approach was incorrect,
2) Integrating du/u, as it is not in the Wolfram Alpha's answer.

In conclsion, it seems, that either my whole approach is incorrect, which is unlikely (as changing the variables and the algebra seems to be OK), or I am making some other fundamental mistakes I can't really see right now.

I'd really appreciate if someone could answer. Thank you.

[EdEarl]

My math is too rusty to help you, sorry. I'm not sure why no one replied to your post, because there are so really good mathematicians on this forum. If you don't get a reply soon, you might try posting your work using LaTeX, See:

http://www.math.harvard.edu/texman/

http://mirrors.ibiblio.org/CTAN/info/symbols/comprehensive/symbols-a4.pdf

http://web.ift.uib.no/Teori/KURS/WRK/TeX/symALL.html

 

For example: [math]\sqrt{\frac{a^2+b^2}{ab^2}}[/math]

[xia]

My math is too rusty to help you, sorry. I'm not sure why no one replied to your post, because there are so really good mathematicians on this forum. If you don't get a reply soon, you might try posting your work using LaTeX, See:

http://www.math.harvard.edu/texman/

http://mirrors.ibiblio.org/CTAN/info/symbols/comprehensive/symbols-a4.pdf

http://web.ift.uib.no/Teori/KURS/WRK/TeX/symALL.html

 

For example: [math]\sqrt{\frac{a^2+b^2}{ab^2}}[/math]

Thank you! I will use it next time.. IHopefully someone will answer soon.