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milkacha

  • 2 years ago

integral of x^2*sqrt(1-x^2) dx using trig substitution

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  1. hartnn
    • 2 years ago
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    did u try x = sin u ??

  2. hartnn
    • 2 years ago
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    then you can always write sin^2 u cos^2 u du as 1/4 (sin^2 2u)du and integrating square of sin function is a pretty standard procedure. let me know if you get stuck in any step.

  3. milkacha
    • 2 years ago
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    I am up to sin^2 u cos^2 u once I rewrite cos^2 u in format 1-sin^2u I get sin^2u - sin^4 u du Now I am stuck.

  4. ganeshie8
    • 2 years ago
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    see if below helps :- sin^2u cos^2u = 1/4 [ sin(2u) ]^2

  5. milkacha
    • 2 years ago
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    Ok so you used a double-angle formula for sin2u. so now I have to simplift 1/4(sin2u)^2? Trig is my weakness - it's like Chinese :(

  6. milkacha
    • 2 years ago
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    Ah yes, I am with you - the product formula!

  7. milkacha
    • 2 years ago
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    Yes so to integrate that I get (1-cos4x)/2 dx using the sine half-angle formula. So I have the integral 1/4(1-cos4x / 2)dx I can replace the 1-cos4x with sin4x so then I have 1/4(sin4x/2)dx Where to from here? :/

  8. milkacha
    • 2 years ago
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    And also I didn't see anywhere where I got rid of the dx = cos x du so there must be an error somewhere

  9. mathmale
    • 2 years ago
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    \[\int\limits_{-}^{-}x ^{2}\sqrt{1-x ^{2}}dx\] let x=1 sin x Then: \[dx=\cos \theta d \theta; x ^{2}=\sin ^{2}\theta; 1-x ^{2}=\cos ^{2}\theta; \sqrt{\cos ^{2}\theta}=\cos \theta\] Then the original integral becomes \[\int\limits_{-}^{-}\sin ^{2}\theta*\cos \theta d \theta=\frac{ \sin ^{3}\theta }{ 3 }+C\]

  10. mathmale
    • 2 years ago
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    Since x=sin theta, x/1=sin theta, and theta = \[\theta=\sin ^{-1}\frac{ x }{ 1 }.\] Substitute this into the prior expression to obtain the integral in terms of x. (Result is mighty simple.)

  11. milkacha
    • 2 years ago
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    Ok thanks I will that way.

  12. milkacha
    • 2 years ago
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    I am still stuck on this one. The answer is: $\frac{x\sqrt{1-x^2}\left(2x^2-1\right)+\arcsin \left(x\right)}{8}+C$ I have NO idea how 1/4(sin^2 2theta) translates into this...... please help!!

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