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Thread: Integral problem (pls help)

  1. #1
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    Integral problem (pls help)

    Does anyone know how to solve this? I can't find similar problems anywhere

    Integral problem (pls help)-screen-shot-2017-11-29-3.29.01-pm.png
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  2. #2
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    Re: Integral problem (pls help)

    By definition of the Sine Integral:

    \displaystyle \dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{d}{dx}\int_0^x \dfrac{\sin \left(t^6\right)}{t^6}dt

    By the Fundamental Theorem of Calculus (sometimes called Part I):

    \displaystyle \dfrac{d}{dx}\int_0^x \dfrac{\sin \left(t^6\right)}{t^6}dt = \dfrac{\sin(x^6)}{x^6}

    This is simply a direct application of the theorem. There is no calculation necessary. Just apply the Fundamental Theorem of Calculus and the answer is immediately apparent.
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    Re: Integral problem (pls help)

    Quote Originally Posted by SlipEternal View Post
    By definition of the Sine Integral:

    \displaystyle \dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{d}{dx}\int_0^x \dfrac{\sin \left(t^6\right)}{t^6}dt

    By the Fundamental Theorem of Calculus (sometimes called Part I):

    \displaystyle \dfrac{d}{dx}\int_0^x \dfrac{\sin \left(t^6\right)}{t^6}dt = \dfrac{\sin(x^6)}{x^6}

    This is simply a direct application of the theorem. There is no calculation necessary. Just apply the Fundamental Theorem of Calculus and the answer is immediately apparent.
    I typed the exact same answer, but it was marked as incorrect. Thank you though!
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  4. #4
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    Re: Integral problem (pls help)

    Ok, I am not familiar with the Si function. I looked it up, but apparently do not understand it. According to Wolframalpha,

    \dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{6\sin(x^6)}{x}

    But, I have no clue how.
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  5. #5
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    Re: Integral problem (pls help)

    Oh, I just misread the description:

    \displaystyle \text{Si}(x^6) = \int_0^{x^6}\dfrac{\sin t}{t} dt

    So, by the Chain Rule, we have:

    \dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{d}{d(x^6)}\left[\text{Si}(x^6)\right]\dfrac{d(x^6)}{dx} = \dfrac{\sin(x^6)}{x^{\cancel{6}}}(6\cancel{x^5}) = \dfrac{6\sin(x^6)}{x}
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  6. #6
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    Re: Integral problem (pls help)

    Quote Originally Posted by SlipEternal View Post
    Ok, I am not familiar with the Si function. I looked it up, but apparently do not understand it. According to Wolframalpha,

    \dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{6\sin(x^6)}{x}

    But, I have no clue how.
    thanks a lot!
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