Integrating a Definite Integral with Trigonometric Functions

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Homework Help Overview

The discussion revolves around evaluating a definite integral involving trigonometric functions, specifically the integral ∫dt/(t^2 + 2t cos a + 1) with limits from 0 to 1, where 0 < a < π. Participants are exploring different substitution methods to simplify the integral.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • One participant attempts a substitution with t = sin a and expresses confusion about the next steps after transforming the integral. Another participant questions the manipulation of csc(a) outside the integral, suggesting it should remain within. A third participant proposes an alternative substitution of t = cos(a) to simplify the integral further.

Discussion Status

The discussion is ongoing, with participants sharing their attempts and questioning each other's reasoning. Some guidance has been offered regarding substitution methods, but there is no explicit consensus on the best approach yet.

Contextual Notes

Participants are working under the constraints of the integral's limits and the condition that 0 < a < π. There is also a note of confusion regarding the proper handling of trigonometric identities and substitutions in the context of the integral.

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Homework Statement


∫dt/(t^2 +2tcos a + 1)
(Limits of the integral are from 0 to 1)
(0<a<π)

Homework Equations

The Attempt at a Solution


Put t=sin a
dt=cosa da
∫dt/(t^2 +2tcos a + 1) = ∫cos a da/(sin^2 a + sin 2a + 1) [ limits of integration changed to 0 to π/2]
= ((cosec a)/2) ∫sin 2a da/(sin^2 a + sin 2a + 1)

I couldn't figure out what to do next... Please help!
 
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utkarsh009 said:

Homework Statement


∫dt/(t^2 +2tcos a + 1)
(Limits of the integral are from 0 to 1)
(0<a<π)

Homework Equations

The Attempt at a Solution


Put t=sin a
dt=cosa da
∫dt/(t^2 +2tcos a + 1) = ∫cos a da/(sin^2 a + sin 2a + 1) [ limits of integration changed to 0 to π/2]
= ((cosec a)/2) ∫sin 2a da/(sin^2 a + sin 2a + 1)
I don't know what you did in this last step, but you can't bring csc(a) out as if it were a constant.
utkarsh009 said:
I couldn't figure out what to do next... Please help!
 
Oh yes... By mistake i typed the integration symbol after it. Cosec a should be inside the integral. So, how should i proceed??
 
I would try a different substitution -- let t = cos(a).That should get you an integral that's easier to do.
 
Oh yah... Thanks ... I solved it... But have a look at this file... I couldn't get an answer for this...
New Doc 1_1.jpg
 

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