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 Post subject: problem with definite integralsPosted: Thu, 17 May 2012 23:49:55 UTC
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Joined: Thu, 17 May 2012 23:43:25 UTC
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I met a problem with calculating following integrals:
[0,pi] (x sinx)/(1+(cos x)^2)
[0,pi/2] [sqrt(sinx)]/[sqrt(sinx) + sqrt(cosx)]

if you could give me a tip with whitch I would be able to proceed with I'd be grateful.

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 Post subject: Re: problem with definite integralsPosted: Fri, 18 May 2012 00:26:21 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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poniek wrote:
I met a problem with calculating following integrals:
[0,pi] (x sinx)/(1+(cos x)^2)
[0,pi/2] [sqrt(sinx)]/[sqrt(sinx) + sqrt(cosx)]

if you could give me a tip with whitch I would be able to proceed with I'd be grateful.

For the first one do a u-substitution. For the second I'd say the same, then multiply by the denominator's conjugate.

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 Post subject: Re: problem with definite integralsPosted: Fri, 18 May 2012 06:25:35 UTC
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Joined: Thu, 17 May 2012 23:43:25 UTC
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But have you done it? I've already tried with the substitution method but as there is also x witch must be changed into arc function it seems to doesn't work...

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 Post subject: Re: problem with definite integralsPosted: Fri, 18 May 2012 06:43:06 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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poniek wrote:
I met a problem with calculating following integrals:
[0,pi] (x sinx)/(1+(cos x)^2)
[0,pi/2] [sqrt(sinx)]/[sqrt(sinx) + sqrt(cosx)]

if you could give me a tip with whitch I would be able to proceed with I'd be grateful.

For the first one do a u-substitution. For the second I'd say the same, then multiply by the denominator's conjugate.

No, there is no need to rationalise the denominator in the second one if one notes the symmetry. For the first one, just use integration by parts and symmetry.

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 Post subject: Re: problem with definite integralsPosted: Fri, 18 May 2012 07:07:22 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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outermeasure wrote:
poniek wrote:
I met a problem with calculating following integrals:
[0,pi] (x sinx)/(1+(cos x)^2)
[0,pi/2] [sqrt(sinx)]/[sqrt(sinx) + sqrt(cosx)]

if you could give me a tip with whitch I would be able to proceed with I'd be grateful.

For the first one do a u-substitution. For the second I'd say the same, then multiply by the denominator's conjugate.

No, there is no need to rationalise the denominator in the second one if one notes the symmetry. For the first one, just use integration by parts and symmetry.

I suppose, I was just thinking for antiderivative's sake.

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 Post subject: Re: problem with definite integralsPosted: Fri, 18 May 2012 07:11:12 UTC
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I suppose, I was just thinking for antiderivative's sake.

They are ugly. The first one is a polylog-type expression, and the second is an elliptic integral.

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 Post subject: Re: problem with definite integralsPosted: Fri, 18 May 2012 08:51:51 UTC
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outermeasure wrote:
I suppose, I was just thinking for antiderivative's sake.

They are ugly. The first one is a polylog-type expression, and the second is an elliptic integral.

Ooh, blech.

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 Post subject: Re: problem with definite integralsPosted: Sun, 20 May 2012 11:01:18 UTC
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Joined: Thu, 17 May 2012 23:43:25 UTC
Posts: 4
SYMMETRY!!!
thanks a million!

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 Post subject: Re: problem with definite integralsPosted: Sun, 20 May 2012 15:10:26 UTC
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Joined: Thu, 17 May 2012 23:43:25 UTC
Posts: 4
For the firs case it works perfectly, but for the second one what substitution have you used? u=tan(x/2) ? - it complicates the integral and doesn't give a clear conclusion. It's also not so easy for u=sinx or u=sqrt(sinx) ....

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 Post subject: Re: problem with definite integralsPosted: Sun, 20 May 2012 15:58:18 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
poniek wrote:
For the firs case it works perfectly, but for the second one what substitution have you used? u=tan(x/2) ? - it complicates the integral and doesn't give a clear conclusion. It's also not so easy for u=sinx or u=sqrt(sinx) ....

.

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