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Annoying trig integral

Hi

I have this integral

\displaystyle\int^\pi_0 \frac{dx}{1 + acos(x)}\ \mathrm{, where\ } a < 1

, the answer to which is

Spoiler

I seem to be useless with the normal tangent half-angle substitution because I can't get my head around the fact that one of my limits becomes undefined. Is there a 'usual' way to deal with this? I have a method to do it by considering the function from 0 to

2\pi

and halving the result, but is there a better way?

(I also suspect the paper meant

|a| < 1

, any thoughts?

*****gr8wizard10*******

(edited 8 years ago)

Reply 1

RichE

Original post by klegend02

Hi

I have this integral

\displaystyle\int^\pi_0 \frac{dx}{1 + acos(x)}\ \mathrm{, where\ } a < 1

, the answer to which is

Spoiler

The t = tan(x/2) substitution should work fine - it transforms the integral into a standard inverse trigonometric one.

I seem to be useless with the normal tangent half-angle substitution because I can't get my head around the fact that one of my limits becomes undefined.

One of the limits becomes +infinity but that is not a problem ultimately, as its arc-tangent is defined.

Is there a 'usual' way to deal with this?

At a more elevated (university) level this is a standard integral to approach with residue theory/complex analysis.

(I also suspect the paper meant

|a| < 1

, any thoughts?

Yes, it should have stated that.

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