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Math Help - Anti derivative

  1. #1
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    Anti derivative

    Confused guys
    Whats the most general anti derivative of

    f(x) = 4x + (3/x^2)

    And

    y(x) = 5x e^(1-x^2) + sin x cos x

    Plz thanks
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  2. #2
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    Re: Anti derivative

    Quote Originally Posted by tankertert View Post
    Confused guys
    Whats the most general anti derivative of

    f(x) = 4x + (3/x^2)

    And

    y(x) = 5x e^(1-x^2) + sin x cos x

    Plz thanks
    For the first one, start by writing \displaystyle \begin{align*} f(x) = 4x + 3x^{-2} \end{align*} and apply the rule \displaystyle \begin{align*} \int{a\,x^n\,dx} = \frac{a\,x^{n+1}}{n+1}  \end{align*}, and for the second

    \displaystyle \begin{align*} \int{5x\,e^{1 - x^2} + \sin{x}\cos{x}\,dx} &= \int{5x\,e^{1 - x^2}\,dx} + \int{\sin{x}\cos{x}\,dx} \\ &= -\frac{5}{2}\int{-2x\,e^{1 - x^2}\,dx} + \int{\sin{x}\cos{x}\,dx} \end{align*}

    Then substitute \displaystyle \begin{align*} u = 1 - x^2 \implies du = -2x\,dx \end{align*} and \displaystyle \begin{align*} v = \sin{x} \implies dv = \cos{x}\,dx \end{align*}
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