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Math Help - Finding antiderivatives

  1. #1
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    Finding antiderivatives

    The antiderivative of the following functions are sought:
    1)(x)^1/2. The solution is, in my booklet, 1/2(x)^1/2
    But if you use the quotient rule ( f(x)/g(x)' => f'(x)g(x)-f(x)g'(x)/ (g(x))^2)
    you get 1/-4x(x)^1/2. Where's the problem? I know, that if you consider 1/(x)^1/2 as x^-1/2 it works out, i just dont see the mistake with applying the quotient rule

    2) 2(x^3)^1/2 Here I'm clueless, how do you find the antiderivative at this example?

    3) 3sin(x). to me the solution is 3(-cos(x)). In my booklet the solution is 3(cos(x)), but isn't the derivative of cos -> -sinus?

    Thanks for you help

    Schdero
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Schdero View Post
    The antiderivative of the following functions are sought:
    1)(x)^1/2. The solution is, in my booklet, 1/2(x)^1/2
    But if you use the quotient rule ( f(x)/g(x)' => f'(x)g(x)-f(x)g'(x)/ (g(x))^2)
    you get 1/-4x(x)^1/2. Where's the problem? I know, that if you consider 1/(x)^1/2 as x^-1/2 it works out, i just dont see the mistake with applying the quotient rule

    2) 2(x^3)^1/2 Here I'm clueless, how do you find the antiderivative at this example?

    3) 3sin(x). to me the solution is 3(-cos(x)). In my booklet the solution is 3(cos(x)), but isn't the derivative of cos -> -sinus?

    Thanks for you help

    Schdero


    Antiderivative means integrating the given function. So, the antiderivative of x^{\frac{1}{2}} would be

    \int x^{\frac{1}{2}} dx =  \frac{2}{3} \times x^{\frac{3}{2}} + C

    If you take the derivative of \frac{2}{3} \times x^{\frac{3}{2}} + C with respect to x, you will get what you integrated, that is, x^{\frac{1}{2}}

    Likewise, the derivative of 3sinx = 3 cos(x), and the antiderivative of 3 sinx = -3cosx+C

    Your answer to find the antiderivative of 3sinx is correct! If what you have asked for (that is finding the antiderivative) is correct, may be your booklet has serious flaws!
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  3. #3
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    Quote Originally Posted by harish21 View Post
    If you take the derivative of \frac{2}{3} \times x^{\frac{3}{2}} + C with respect to x, you will get what you integrated.
    Sorry I dount understand anything of this.
    1)Is \frac{1}{2(x^{0.5})} as antidervative of x^{0.5} henceforth wrong?
    2)why does the quotient rule appereantly not apply here or made i a mistake?
    3) what about 2,3?
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  4. #4
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    sry i cannot edit the last post, please ingore it!
    Thanks for your help!
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