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Substitution and Proving the Identity - HELP

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.
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.


Surely you can just divide by sinx?
Reply 2
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Azland
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Original post by G0TA4s1T1NN3R
Surely you can just divide by sinx?


that would make it Sin x * Cot^2 x right?

Where do I go from there.
(edited 12 years ago)
Original post by Azland
that would make it Sin x * Cot x * Cosx = Cos^2 x right?


Yeah, then you can use D'Jickin's identity to resolve it from there :smile:
Reply 4
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Azland
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Original post by G0TA4s1T1NN3R
Yeah, then you can use D'Jickin's identity to resolve it from there :smile:


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.
Reply 5
Avatar for raheem94
raheem94
13
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.


Why are you using cot here?

y=∫(x21−x2)dx \displaystyle y = \int \left(x^2\sqrt{1-x^2}\right)dx

x=sinθ \displaystyle x=sin\theta

Differentiate the above expression to express, dx, in terms of , dθ \displaystyle d\theta

In your integral, remember to use the identity, cos2x=1−sin2x \displaystyle cos^2x=1-sin^2x
Reply 6
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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 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

:smile:
(edited 12 years ago)
Reply 7
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Azland
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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

:smile:


I reached this point myself initially but then got stuck.

2sin^2Өcos^2Ө = sin2Ө

Lost you here. Isnt Sin2Ө = 2SinӨCosӨ
(edited 12 years ago)
Reply 8
Avatar for raheem94
raheem94
13
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

:smile:


Original post by Azland
2sin^2Өcos^2Ө = sin2Ө

Lost you here. Isnt Sin2Ө = 2SinӨCosӨ


sin2θ=2sinθcosθ[br]sin22θ=4sin2θcos2θ \displaystyle sin2\theta = 2sin\theta cos\theta [br]\displaystyle sin^22\theta = 4sin^2\theta cos^2\theta

sin22θ4=sin2θcos2θ \displaystyle \frac{sin^22\theta}{4} = sin^2\theta cos^2\theta
(edited 12 years ago)
Reply 9
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Azland
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Got it, thanks. !
Reply 10
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Pin
Yep, you're right. Wrote down the right working on my piece of paper :/
glad you still got it. :smile:
(edited 12 years ago)

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