How can I evaluate this integral using trig substitution?

In summary, to evaluate the integral of 1/(x^2 sqrt(x^2 - 9)) using trig substitution, you can use x = 3sec(theta) and create a right triangle with 3 on the hypotenuse, x on one leg, and sqrt(x^2 - 9) on the other. Then, using the identity (sec(theta))^2 - 1 = tan(theta)^2, you can substitute for dx/d(theta) and solve the integral.
  • #1
montana111
12
0

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.
 
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  • #2
The hint is infact very useful.

you have:

[tex]\int\frac{1}{x^2\sqrt{x^2-9}}dx[/tex]

using the substitution x = 3sec[tex]\theta[/tex]
you will get a simple integral.

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

so after the substitution, what do you get?
(You might like to remember also the identity (sec[tex]\theta[/tex])^2-1=tan([tex]\theta[/tex])^2)
:wink:
 
  • #3
montana111 said:
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.
 

1. What is trig substitution?

Trig substitution is a method used to solve integrals that involve expressions of the form sqrt(a^2-x^2), sqrt(x^2-a^2), or sqrt(a^2+x^2), where a is a constant. It involves substituting a trigonometric function for x in order to simplify the integral.

2. When should I use trig substitution?

Trig substitution is most commonly used when solving integrals that involve expressions of the form sqrt(a^2-x^2), sqrt(x^2-a^2), or sqrt(a^2+x^2). It can also be used to simplify integrals involving rational expressions or logarithmic functions.

3. What are the three main types of trig substitution?

The three main types of trig substitution are sine substitution, cosine substitution, and tangent substitution. These substitutions involve replacing x with either sinθ, cosθ, or tanθ, respectively, and then using trigonometric identities to simplify the integral.

4. How do I know which trig substitution to use?

The type of trig substitution to use depends on the expression being integrated. If the expression contains sqrt(a^2-x^2), then the sine substitution should be used. If it contains sqrt(x^2-a^2), the cosine substitution is appropriate. And if it contains sqrt(a^2+x^2), the tangent substitution is the best choice.

5. What are some tips for using trig substitution effectively?

When using trig substitution, it is important to choose the substitution that will make the integral as simple as possible. This may require using trigonometric identities, simplifying fractions, or manipulating the integral to fit a specific form. It is also important to carefully keep track of any substitutions made and to substitute back in for θ at the end of the integration process.

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