How can I evaluate this integral using trig substitution?

[montana111]

Homework Statement

evaluate the integral using the indicated trig substitution. sketch the corresponding right triangle.

integral of(1/(x^2 sqrt(x^2 - 9))

Homework Equations

integral of(1/(x^2 sqrt(x^2 - 9))

The Attempt at a Solution

at first glance this seemed really easy, and i tried doing it without the given trig substitution of 3sec(theta), but i just got confused. i made the triangle with 3 on the hypotenuse and x on the leg opposite theta. then i said x = 3sin(theta) (because i have a value for the hypotenuse and the opposite) and from there i have no idea what to do. Also, I am discouraged because the book tells you to use 3sec(theta) which is driving me insane because i have no idea why i would use sec at all. thanks for you help.

 

[sara_87]

The hint is infact very useful.

you have:

$$\int\frac{1}{x^2\sqrt{x^2-9}}dx$$

using the substitution x = 3sec$$\theta$$
you will get a simple integral.

substitute this into the integral. BUT... you have to first find dx/d$$\theta$$ in order to substitute something for dx (in terms of d$$\theta$$)

so after the substitution, what do you get?
(You might like to remember also the identity (sec$$\theta$$)^2-1=tan($$\theta$$)^2)

 

[Mark44]

i made the triangle with 3 on the hypotenuse and x on the leg opposite theta.

Since the radical contains x² - 9, you want x on the hypotenuse and 3 on one leg, and sqrt(x² - 9) on the other leg.