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Math Help - Trignometric Integrals

  1. #1
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    Trignometric Integrals

    <br />
\int \frac{x^2}{\sqrt{4-x^2}} dx<br /> <br />

    <br /> <br />
\int sin^2(x)cos^3(x) dx<br /> <br />

    I cant figure these 2 out.
    Can you also show the steps?
    Thank You so much
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  2. #2
    zeg
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    I would use u substitution on the first one using:
    u = 4-x^2

    and for the second one, try integration by parts:
    \int udv = uv - \int vdu
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  3. #3
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    Quote Originally Posted by larryboi7 View Post
    <br />
\int \frac{x^2}{\sqrt{4-x^2}} dx<br /> <br />

    <br /> <br />
\int sin^2(x)cos^3(x) dx<br /> <br />

    I cant figure these 2 out.
    Can you also show the steps?
    Thank You so much
    use substitution method

    HINT :
    1) let x= 2 sin t
    use sin^2(x)=1-cos^2(x)
    2) <br /> <br />
\int sin^2(x)cos^3(x) dx =\int sin^2(x) cos^2(x) cos(x)dx <br /> <br />
    use cos^2(x)=1-sin^2(x) and let sin x=t so that cosx dx =dt and simplify
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  4. #4
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    1)

    x=2 \sin t \rightarrow dx=2 \cos t dt
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  5. #5
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    how do i find sin2t?
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  6. #6
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    For the 2nd one, use cos(x)^2=1 - sin(x)^2
    Then let u = sin x, and the remaining cos x is the du.
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  7. #7
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    Quote Originally Posted by danielomalmsteen View Post
    1)

    x=2 \sin t \rightarrow dx=2 \cos t dt
    after doing all that, how do i solve for sin2t?
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  8. #8
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    Quote Originally Posted by larryboi7 View Post
    after doing all that, how do i solve for sin2t?
    to restore the changes of variables?
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  9. #9
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    Quote Originally Posted by danielomalmsteen View Post
    to restore the changes of variables?
    yes
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  10. #10
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    \sin {2t} = 2 \cdot \sin t \cdot \cos t = 2x \cdot \sqrt{1- \sin^2 t} = 2x \cdot \sqrt{1-\frac{x^2}{4}} = x \cdot \sqrt{4-x^2}
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