Question

Evaluate the integral: ∫(x^4 + x - 4)/(x^2 +2)
I've been stuck on this problem for the longest time. Any explanation would be greatly appreciated! :)

Answer 1

I got you.

Answer 2

First thing to notice is that:
\[\int\limits \frac{x^4}{x^2+2}dx + \int\limits \frac{xdx}{x^2+2} - \int\limits \frac{4}{x^2+2}dx\]

Answer 3

So looking at the **second** integral. We see this is a simply u-sub; u=x^2+2 implies du = 2x dx, easy.
The last integral is done by noting that:
\[\int\limits \frac{dx}{x^2+a^2}=\frac{1}{a}\arctan(\frac{x}{a})\]
Here, a = sqrt(2).

Answer 4

Ahh, I see now. Thank you so much for your help, I was going insane trying to work this one out.

Answer 5

Actually, for the first integral I should have went with my original instincts. Long division. Do that and then it'll be in a nice form!

Answer 6

I didn't think of that! Thanks a million. :)

Answer 7

\[\implies \int\limits (x^2-2)dx+\int\limits \frac{x dx}{x^2+2}=\frac{x^3}{3}-2x+\frac{1}{2}\ln|x^2+2|+C\]

Answer 8

I finished it for you :P Nice trick.

Answer 9

I just like being explicit :P