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Math Help - Integrals

  1. #1
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    Integrals

    Hello!

    I'm having some problems with Calculus. I can't even figure out how to get started, let alone do the problem :S



    1)Find derivative of the function:




    2)Calculate area enclosed by y^2 = 2x+6 and y = x -1


    3) Calculate the integral
    <br />
\int \frac {\sin2x}{1+cos^2x} dx<br />


    Thanks guys
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  2. #2
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    Quote Originally Posted by funnytim View Post
    Hello!

    I'm having some problems with Calculus. I can't even figure out how to get started, let alone do the problem :S



    1)Find derivative of the function:




    2)Calculate area enclosed by y^2 = 2x+6 and y = x -1


    3) Calculate the integral
    <br />
\int \frac {\sin2x}{1+cos^2x} dx<br />


    Thanks guys

    I'll give you a hand with the first one, you try to make some work of your own with the other ones:

    Put F(x):=\int\limits_{\tan x}^{x^2}\frac{1}{\sqrt{2+t^4}}dt As the function in the integral is continuous and defined everywhere, the Fundamental Theorem of integral Calculus tells us that

    F(x)=G(x^2)-G(\tan x) , where G is a primitive function of the integrand function, but then:

    F'(x)=2xG'(x^2)-\frac{1}{\cos^2x}G'(\tan x) =2x\,\frac{1}{\sqrt{2+x^8}}-\frac{1}{\cos^2x}\,\frac{1}{\sqrt{2+\tan^4x}}

    Tonio
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  3. #3
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    For question three .

    Consider  \sin(2x) = 2\sin(x) \cos(x)

    Then make a suitable substitution
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  4. #4
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    1.For your first question see attachment


    2. For your second question see second attachment


    3.For your third question write sin(2x) = 2sin(x)cos(x)

    Then make the substitution u = cos(x) you will obtain -2u/(1+u^2)

    in the integrand which you should be able to do
    Attached Thumbnails Attached Thumbnails Integrals-2dftc.jpg   Integrals-integral.jpg  
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  5. #5
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    Thanks guys, you're great!

    One question in#2 though. After i obtain:




    How do I calculate it to obtain the answer, 18?

    Thank you again!
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  6. #6
    MHF Contributor Calculus26's Avatar
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    Integrate and evaluate at the limits---submit your work if you don't get 18 and I'll see what happened
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  7. #7
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    Actually, while I'm puzzling over that problem, here's another: How do I find a definite integral?

    In particular:

    Thanks again.
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  8. #8
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    Quote Originally Posted by funnytim View Post
    Actually, while I'm puzzling over that problem, here's another: How do I find a definite integral?

    In particular:

    Thanks again.

    Doesn't it look suspiciously similar to a Riemann sum of the function \frac{x}{x^2+1} over the interval [0,1]?

    Tonio
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