Question

integrate x^3/(x^2-9) integrate trig substitution

Answer 1

trig sub? oo these are fun :D

Answer 2

\[\large \int\limits\frac{x^3}{x^2-9}dx\]

This is of the form \(\large x^2-a^2\)
So we'll want to let \(\large x=a\sec\theta\)

\[\large x^2-a^2 \qquad=\qquad (a \sec \theta)^2-a^2\qquad=\qquad a^2(\sec^2\theta-1)\]\[\large =a^2 \tan^2 \theta\]

Understand the substitution? :o

Answer 3

yes i understand that is x=3sec(theta) x^2=9sec^2(theta)
dx=3sec(theta)tan(theta)

Answer 4

\[\int\limits_{?}^{?}(3\sec(\theta) ^33\sec(\theta)\tan(\theta)d(\theta))/ (9\sec^2(\theta)-9)\]

Answer 5

yah looks good so far :)

Answer 6

I wanna rewrite that, it's just a little too sloppy for my liking lol.\[\large \int\limits \frac{(3\sec \theta)^3 \left[3\sec \theta \tan \theta \;d \theta\right]}{9\sec^2\theta-9}\]

Answer 7

Looks like we have a lot of nice cancellations :o

Answer 8

\[\int\limits_{?}^{?} (3\sec(\theta))^33\sec(\theta)\tan(\theta)d(\theta)/9\sec^2(\theta)-9

Answer 9

\[then \because \sec^2(\theta)-1=\tan^2(\theta), then 9\int\limits\limits_{}^{?}\sec^4(\theta)\tan(\theta)/\tan(\theta)\]\]

Answer 10

that last tan is suppose be squared

Answer 11

kk, good sounds right.

Answer 12

\[then \because \sec^2(\theta)-1=\tan^2(\theta), then 9\int\limits\limits_{}^{?}\sec^4(\theta)\tan(\theta)/\tan^2(\theta)\]

Answer 13

then the tan cancelled out and then im left with\[9\int\limits \sec^4(\theta)d(\theta)/ \tan(\theta)\] then i dont know what to do

Answer 14

hmmm ok, sec thinking.

Answer 15

Oh ok, I guess the next step would be to write the top of our fraction in terms of tangents.

Answer 16

\[\large 9\int\limits \frac{\sec^2\theta \sec^2\theta}{\tan \theta}d \theta\]
Then we apply our square identity for sec^2 to change it to tangents maybe? Hmmm I think that will work...

Answer 17

Yah this seems to be working.
You understand what I'm saying? :o

After you use your square identity, multiply out the top.
Then you can break it up into a bunch of easier fractions.

Answer 18

Need to see steps?

Answer 19

i think i understand thank you

Answer 20

ok lemme know if you're stuck c:

Answer 21

I saw trig sub and was excited and then noticed you had already took the fun. Haha