Results are out! Find what you need...fast. Get quick advice or join the chat
Hey! Sign in to get help with your study questionsNew here? Join for free to post

Substitution and Proving the Identity - HELP

Announcements Posted on
Will you get the grades you need for uni? Get prepared today and de-stress, sign up to email alerts for course places! 02-06-2015
Waiting on IB results? Our IB results hub explains everything you need to know 01-06-2015
  1. Offline

    ReputationRep:
    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.
  2. Offline

    ReputationRep:
    (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?
  3. Offline

    ReputationRep:
    (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.
  4. Offline

    ReputationRep:
    (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
  5. Offline

    ReputationRep:
    (Original post by G0TA4s1T1NN3R)
    Yeah, then you can use D'Jickin's identity to resolve it from there
    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.
  6. Offline

    ReputationRep:
    (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?

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

     \displaystyle x=sin\theta

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

    In your integral, remember to use the identity,  \displaystyle  cos^2x=1-sin^2x
  7. Offline

    ReputationRep:
    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

  8. Offline

    ReputationRep:
    (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

    I reached this point myself initially but then got stuck.

    2sin^2Өcos^2Ө = sin2Ө

    Lost you here. Isnt Sin2Ө = 2SinӨCosӨ
  9. Offline

    ReputationRep:
    (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

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

    Lost you here. Isnt Sin2Ө = 2SinӨCosӨ
     \displaystyle sin2\theta = 2sin\theta cos\theta 

\displaystyle sin^22\theta = 4sin^2\theta cos^2\theta

     \displaystyle \frac{sin^22\theta}{4} = sin^2\theta cos^2\theta
  10. Offline

    ReputationRep:
    Got it, thanks. !
  11. Offline

    ReputationRep:
    Yep, you're right. Wrote down the right working on my piece of paper :/
    glad you still got it.

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By joining you agree to our Ts and Cs, privacy policy and site rules

  2. Slide to join now Processing…

Updated: April 3, 2012
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

New on TSR

It's IB results day tomorrow

Check out our IB results hub here & get prepared here

Study resources
x

Think you'll be in clearing or adjustment?

Hear direct from unis that want to talk to you

Get email alerts for university course places that match your subjects and grades. Just let us know what you're studying.

Quick reply
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.