Difficult integral for Trig Substitution

In summary, the problem has the user frustrated because the derivation is not at all clear. They have tried integration by parts, but every approach ended with an even more complicated integral. They have found that trigonometric substitution is the best option, but are looking for help.
  • #1
bigred09
3
0
ok i have been studying the in-depth processes of trigonometric substitution with integrals and this problem has me frusterated.

[tex]\int x^2\sqrt{(x^2-4)} dx[/tex]

The evaluation is clear (from an old Table of Integrals I found), but the derivation is not at all clear, which is what i want to know.

I also tried to solve this by integration by parts, but every approach ended with an even more complicated integral, so trig substitution is probably the best choice.

Can anyone help?
 
Physics news on Phys.org
  • #2
Try the substitution u = 2sec(theta)
 
  • #3
bigred09 said:
ok i have been studying the in-depth processes of trigonometric substitution with integrals and this problem has me frusterated.

[tex]\int x^2\sqrt{(x^2-4)} dx[/tex]
[itex]sin^2(\theta)+ cos^2(\theta)= 1[/itex] so [itex]sin^2(\theta)= 1- cos^2(\theta)[/itex] and, dividing on both sides by [itex]cos^2(\theta)[/itex], [itex]tan^2(\theta)= sec^2(\theta)- 1[/itex]. The substitution [itex]x= 2sec(\theta)[/itex], as bigred09 suggested, will reduce that squareroot to [itex]2 tan(\theta)[/itex].

The evaluation is clear (from an old Table of Integrals I found), but the derivation is not at all clear, which is what i want to know.

I also tried to solve this by integration by parts, but every approach ended with an even more complicated integral, so trig substitution is probably the best choice.

Can anyone help?
 
  • #4
Alternatively, use the hyperbolic substitution [tex]x=2Cosh(u)[/tex]
 
  • #5
ok well with trig substitution, i get

[tex]\int tan^2\theta sec^3\theta d\theta[/tex]

which doesn't help me. Can somone solve this integral then?
 
  • #6
bigred09 said:
ok well with trig substitution, i get

[tex]\int tan^2\theta sec^3\theta d\theta[/tex]

which doesn't help me. Can somone solve this integral then?

Wrong! Look what Halls said, post #3.
 
  • #7
Letting

[tex]x=2sec(\theta)=>4sec^2(\theta)\sqrt{4(sec^2(\theta)-1)}=4sec^2(\theta)*2\sqrt{tan^2(\theta)}=...{[/tex]
Edit: Disregard this!
 
Last edited:
  • #8
@ sutupidmath:

You're forgetting about dx/d(theta)
 
  • #9
JG89 said:
@ sutupidmath:

You're forgetting about dx/d(theta)

:redface:
 
  • #10
right i actually forgot the coefficient 8 but that doesn't mess with the integral. and [tex]dx=sec\theta tan\theta[/tex]


so what halls said wass valid. all i did was simplify it more. even more so it looks like this:

[tex]8\int \frac{cos^5\theta}{sin^3\theta} d\theta[/tex]


so uh...seriously...any ideas on solving this?
 

1. What is a difficult integral for Trig Substitution?

A difficult integral for Trig Substitution is an integral that involves trigonometric functions, such as sine, cosine, and tangent, that cannot be solved using basic integration techniques. These integrals require the use of Trig Substitution, a method that involves substituting a trigonometric function for a variable in the integral to make it solvable.

2. Why is Trig Substitution necessary for difficult integrals?

Trig Substitution is necessary for difficult integrals because it allows us to rewrite the integral in terms of trigonometric functions, which are easier to integrate. This method is particularly useful for integrals involving square roots, rational functions, and certain combinations of trigonometric functions.

3. What are the steps for solving a difficult integral using Trig Substitution?

The steps for solving a difficult integral using Trig Substitution are:
1. Identify the integral as a difficult one that involves trigonometric functions.
2. Determine which trigonometric substitution to use based on the form of the integral.
3. Substitute the trigonometric function for the variable in the integral.
4. Simplify the integral using trigonometric identities.
5. Solve the resulting integral using basic integration techniques.
6. Substitute the original variable back into the solution.

4. What are some common trigonometric substitutions used in difficult integrals?

Some common trigonometric substitutions used in difficult integrals are:
- For integrals involving √(a^2 - x^2), use x = a sinθ
- For integrals involving √(a^2 + x^2), use x = a tanθ
- For integrals involving √(x^2 - a^2), use x = a secθ
- For integrals involving √(x^2 + a^2), use x = a cotθ
These substitutions are chosen based on the Pythagorean identities and the reciprocal identities of trigonometric functions.

5. Are there any tips for solving difficult integrals using Trig Substitution?

Yes, here are some tips for solving difficult integrals using Trig Substitution:
- Always check to see if the integral can be simplified using algebraic methods before attempting Trig Substitution.
- Be familiar with the various trigonometric identities to simplify the integral as much as possible.
- Pay attention to the limits of integration when substituting back to the original variable.
- Practice and familiarize yourself with different types of difficult integrals to improve your skills in solving them.

Similar threads

  • Calculus
Replies
6
Views
1K
Replies
3
Views
934
  • Calculus
Replies
9
Views
776
Replies
14
Views
1K
  • Calculus
Replies
29
Views
513
  • Calculus
Replies
1
Views
1K
Replies
4
Views
187
Replies
13
Views
926
Replies
4
Views
1K
Back
Top