Trig Integration Help: What & How to Use Trig Substitutions

In summary, trigonometric substitution involves making a substitution using a trigonometric function, such as sin(u) or tan(u), to simplify an integral. After integrating, the inverse substitution is made to get the final answer. It is a useful technique in solving integrals, as shown in the sample problem provided.
  • #1
expscv
241
0
help wat is mean by trigonometric substituations? how do i use it/? thx
 
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  • #2
hah, another part of calc 2 that I've sort of forgotten.., let me get out my calc book and ill get back :)
 
  • #3
You make a substitution of the form x = some trig function (like sin(u) or tan(u)) and then use trig properties to simplify the integral to a manageable form. Then, after integrating, you make the inverse substitution into the solution to get your final answer.

Look at the table at the bottom of Mathworld's entry and just make an up example.

Trigonometric Substitution:
http://mathworld.wolfram.com/TrigonometricSubstitution.html

Better yet, here's a sample problem for you. Try it out.

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

cookiemonster
 
  • #4
Figured I should work out my own sample problem.

[tex]\int \frac{dx}{\sqrt{a^2 - x^2}}[/tex]
[tex]x = a\sin{\theta}[/tex]
[tex]dx = a\cos{\theta}d\theta[/tex]
[tex]\int\frac{a\cos{\theta}d\theta}{\sqrt{a^2(1-\sin^2{\theta})}}[/tex]
[tex]\int\frac{a\cos{\theta}d\theta}{a\cos{\theta}}[/tex]
[tex]\int d \theta[/tex]
[tex]\theta = \arcsin{\frac{x}{a}}[/tex]

cookiemonster
 
  • #5
i do remember that
[tex]sin^2 + cos^2 = 1 [/tex]

so there for, you can rearrange...

... wait, I am thinking of trig identities.

hrm, maybe i shouldn't be doing this calc stuff... i have too many brain farts.
 
  • #6
Trig identities are good. Keep going.

You'll notice that that particular trig identity is used to get from the 4th step in my example to the 5th step.

cookiemonster
 
  • #7
oh cool thx
this was my problem in solving it

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

[tex] x=tan{\theta} [/tex]
my step after ur help

[tex] \int \frac {1}{tan^2{\theta}sec{\theta}}dx [/tex]

[tex] x=tan{\theta} [/tex]then [tex] dx=sec^2{\theta} d{\theta}[/tex]

[tex] \int \frac {sec^2{\theta}}{Tan^2{\theta}sec{\theta}}d{\theta}[/tex]

[tex]\int \frac {sec{\theta}}{tan^2{\theta}} d{\theta} [/tex]

[tex]\frac{1}{2}\int \frac {2sec{\theta}}{1-sec^2{\theta}} d{\theta} [/tex]


is this correct ? thx
 
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  • #8
? wat happen to my latex graphic?
 
  • #9
You're working too hard.

Try to simplify

[tex]\int \frac {sec{\theta}}{tan^2{\theta}} d{\theta} [/tex]

a bit. Once you do, there's an obvious substitution that completes the integral.

cookiemonster
 
  • #11
What do you mean you can't simplify anymore?

[tex]\int \frac{\sec{\theta}}{\tan^2{\theta}} \, d\theta = \int \frac{1}{\cos{\theta}} \cdot \frac{\cos^2{\theta}}{\sin^2{\theta}} \, d\theta = \int \frac{\cos{\theta}}{\sin^2{\theta}} \, d\theta[/tex]

There's a substitution that let's you evaluate this integral...

And which thread are you referring to?

cookiemonster

Edit: Gaah! Why won't LaTeX work?

Edit^2: Guess it does work.
 
Last edited:
  • #13
That keeps going to the index of the General Math forum, not a specific thread. Which specific thread in General Math?

cookiemonster
 

1. What is a trig substitution?

A trig substitution is a method used to solve integrals involving trigonometric functions. It involves replacing a complicated expression in the integrand with a simpler trigonometric expression, making it easier to solve the integral.

2. When should I use trig substitutions?

Trig substitutions are useful when the integrand contains a radical with a sum or difference of squares, or when the integrand contains expressions involving the trigonometric functions sine, cosine, tangent, or secant.

3. How do I use a trig substitution?

To use a trig substitution, first identify the trigonometric function in the integrand that can be substituted. Then, substitute the corresponding trigonometric expression (such as sine or tangent) for that function. Finally, use trigonometric identities to simplify the integral and solve for the variables.

4. Are there any common trigonometric identities that can help with trig substitutions?

Yes, there are several common trigonometric identities that can be used when using trig substitutions, such as the Pythagorean identities, the double angle formulas, and the sum and difference identities.

5. Can I use trig substitutions in all integrals?

No, trig substitutions are only useful in specific cases where the integrand contains certain expressions involving trigonometric functions. In some cases, other integration techniques such as integration by parts or partial fractions may be more effective.

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