Question

By completing the square and making a substitution, evaluate:

Answer 1

\[\int\limits_{}^{} \frac{ dx }{ x ^{2}+x+1 }\]

Answer 2

I think I completed the square right but I'm not sure how exactly this process works or really applies to solving it all that much.

Answer 3

what is the denominator of the integrand when you completed the square?

Answer 4

I'm shaky with completing the square but I got (x+(1/2))^2 +(3/4)

Answer 5

I wasn't exactly sure what to do with the numbers you'd normally add to the other side, as there was no other side...

Answer 6

yep... that's correct..
now isn't that of the form \(\large \frac{1}{x^2+a^2} \) ????

Answer 7

For some reason our teacher decided to leave us in the dark about these things....it would seem so but I wasn't sure if that form needed perfect squares...

Answer 8

Also our book decided to put this problem in the section right before that comes up for some reason. Umm...the answer has inverse tangent in it, is that what you get from that form?

Answer 9

you mean for the "a" part? no... don't need perfect squares...

Answer 10

right and ok cool, the example in my book was x^2+4 so i wasn't sure

Answer 11

What do I do with the 3/4 though? Can I take that out and integrate it separately?

Answer 12

so that integral is arctan (if i recall correctly...)

Answer 13

\(\large \frac{3}{4}=(\frac{\sqrt3}{2})^2 \)

Answer 14

Now I'm confused.

Answer 15

Oh, (x+1/2)^2 is the x^2 and 3/4 is the a^2?

Answer 16

your integrand can be written in the form \(\large \frac{1}{u^2+a^2} \)
where \(\large u=x+\frac{1}{2} \) and \(\large a=\frac{\sqrt3}{2} \)

Answer 17

yes....

Answer 18

I was looking at it differently, ok that makes more sense. And it doesn't matter which is a and which is x right? I mean obviously I could switch the order right?

Answer 19

correct..

Answer 20

Ok, thanks so much!!

Answer 21

yw... :)