Question

integral 1/(sqroot(e^(2x)-1)) dx
solve using trig substitution.

I can solve it using other methods, but I can't using trig substitution...

Answer 1

\[\Large\bf\sf \int\limits \frac{1}{\sqrt{e^{2x}-1}}dx\quad=\quad \int\limits \frac{1}{\sqrt{\left(e^x\right)^2-1}}dx\]Let:\[\Large\bf\sf e^x=\sec \theta\]

Answer 2

multiply both top and bottom by e^x, and then let u = e^x.

Answer 3

Do you understand how I applied the exponent rule and got a square in the bottom?
I guess that's the tricky part.
After you get a square, you can apply a trig sub.

Understand why I chose secant?

Answer 4

I got to that point. However, I got lost at that part. How do we use sec?

Answer 5

@sourwig The teacher asks us to not use that method.

Answer 6

\[\Large\bf\sf \tan^2\theta+1=\sec^2\theta \qquad\to\qquad \color{royalblue}{\sec^2\theta-1=\tan^2 \theta}\]Using secant (and it's square identity) will allow us to get rid of the subtraction under the root.
We'll be able to simplify it from there.

Answer 7

\[\Large\bf\sf \int\limits\limits \frac{1}{\sqrt{\left(\sec \theta\right)^2-1}}dx\quad=\quad \int\limits \frac{1}{\tan \theta}\;dx\]Understand how I did that?
I skipped a couple steps in there.
We just need to deal with the differential dx before we can integrate.

Answer 8

I think I got it. But, please, send what you are about to write. It is always useful.

Answer 9

dx=tanthetadtheta.

Answer 10

integral of 1 dtheta

Answer 11

=theta

Answer 12

=arcsec(e^x)+C

Answer 13

oh boy oh boy i got this :)

Answer 14

Oh good c: