Substitution and Proving the Identity - HELP
Maths and statistics discussion, revision, exam and homework help.
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Substitution and Proving the Identity - HELPy = x^2 √(1 − x^2)
substitution x = sin Ө
Prove that that is
1/4 * sin^2 2Ө dӨ
Got close to solving it once but then messed it up. Please help through steps as I'm getting stuck at one particular point. -
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Re: Substitution and Proving the Identity - HELPSurely you can just divide by sinx?(Original post by Azland)
y = x^2 √(1 − x^2)
substitution x = sin Ө
Prove that that is
1/4 * sin^2 2Ө dӨ
Got close to solving it once but then messed it up. Please help through steps as I'm getting stuck at one particular point. -
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Re: Substitution and Proving the Identity - HELPthat would make it Sin x * Cot^2 x right?(Original post by G0TA4s1T1NN3R)
Surely you can just divide by sinx?
Where do I go from there.Last edited by Azland; 03-04-2012 at 18:31. -
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Re: Substitution and Proving the Identity - HELPYeah, then you can use D'Jickin's identity to resolve it from there(Original post by Azland)
that would make it Sin x * Cot x * Cosx = Cos^2 x right?
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Re: Substitution and Proving the Identity - HELPMy bad in the last one. I forgot to divide the Cos x you get from changing dx to dӨ so its actually Sin x * Cot^2 x(Original post by G0TA4s1T1NN3R)
Yeah, then you can use D'Jickin's identity to resolve it from there
Lost on what to do after that. -
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Re: Substitution and Proving the Identity - HELPWhy are you using cot here?(Original post by Azland)
My bad in the last one. I forgot to divide the Cos x you get from changing dx to dӨ so its actually Sin x * Cot^2 x
Lost on what to do after that.
Differentiate the above expression to express, dx, in terms of ,
In your integral, remember to use the identity,
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Re: Substitution and Proving the Identity - HELPx = sin Ө
dx/dӨ = cos Ө so you can replace dx with cosӨ dӨ
√1-x^2 = √1-sin^2Ө=√cos^2Ө = cos Ө
= sin^2ӨcosӨcosӨ d
=sin^2Өcos^2Ө dӨ
2sin^2Өcos^2Ө = sin2Ө
2sinӨcosӨ = sin2Ө
so if you do sin^2Өcos^2Ө x 4 = 4sin^2Өcos^2Ө
which is equal to 2sinӨcosӨ2sinӨcosӨ = (sin2Ө)^2
as you multiply by 4 you then have to divide by 4
therefore
1/4 sin^2 2Ө
Last edited by Pin; 03-04-2012 at 19:39. Reason: the workings now make sense -
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Re: Substitution and Proving the Identity - HELPI reached this point myself initially but then got stuck.(Original post by Pin)
x = sin Ө
dx/dӨ = cos Ө so you can replace dx with cosӨ dӨ
√1-x^2 = √1-sin^2Ө=√cos^2Ө = cos Ө
= sin^2ӨcosӨcosӨ dӨ
=sin^2Өcos^2Ө dӨ
2sin^2Өcos^2Ө = sin2Ө
so if you do sin^2Өcos^2Ө x 4 = 4sin^2Өcos^2Ө
which is equal to (sin2Ө)^2
as you multiply by 4 you then have to divide by 4
therefore
1/4 sin^2 2Ө
2sin^2Өcos^2Ө = sin2Ө
Lost you here. Isnt Sin2Ө = 2SinӨCosӨLast edited by Azland; 03-04-2012 at 18:51. -
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Re: Substitution and Proving the Identity - HELP(Original post by Pin)
x = sin Ө
dx/dӨ = cos Ө so you can replace dx with cosӨ dӨ
√1-x^2 = √1-sin^2Ө=√cos^2Ө = cos Ө
= sin^2ӨcosӨcosӨ dӨ
=sin^2Өcos^2Ө dӨ
2sin^2Өcos^2Ө = sin2Ө
so if you do sin^2Өcos^2Ө x 4 = 4sin^2Өcos^2Ө
which is equal to (sin2Ө)^2
as you multiply by 4 you then have to divide by 4
therefore
1/4 sin^2 2Ө
Last edited by raheem94; 03-04-2012 at 18:54.
