Trig substitution into integrals


by tadf2
Tags: integral test, integrals, substitution, trig, trig substitution
tadf2
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#1
Feb15-14, 08:32 PM
P: 5
I was testing for convergence of a series:
∑[itex]\frac{1}{n^2 -1}[/itex] from n=3 to infinity

I used the integral test, substituting n as 2sin(u)

so here's the question:
when using the trig substitution, I realized the upperbound, infinity, would fit inside the sine.

Is it still possible to make the substitution? Or is there a restriction when this happens?
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Mark44
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#2
Feb15-14, 09:10 PM
Mentor
P: 21,069
Quote Quote by tadf2 View Post
I was testing for convergence of a series:
∑[itex]\frac{1}{n^2 -1}[/itex] from n=3 to infinity

I used the integral test, substituting n as 2sin(u)

so here's the question:
when using the trig substitution, I realized the upperbound, infinity, would fit inside the sine.
What does "fit inside the sine" mean?
Quote Quote by tadf2 View Post

Is it still possible to make the substitution? Or is there a restriction when this happens?
Sure, you can make the substitution. The integral will be from 3 to, say b, and you take the limit as b → ∞.

Not that you asked, but it's probably simpler and quicker to break up 1/(n2 - 1) using partial fractions.
tadf2
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#3
Feb16-14, 08:29 PM
P: 5
Inside the sine meaning, the argument of the 'arcsine' would only range from -1 to 1.
So I'm guessing you can't make the substitution because arcsin(infinity) = error?

gopher_p
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#4
Feb16-14, 09:01 PM
P: 337

Trig substitution into integrals


If you're looking for an appropriate trig substitution for the definite integral (and not just one that gets you a correct antidierivative), then ##\sec u## is the way to go. But like Mark44 said, partial fractions is really the "right" technique of integration for this particular integral.


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