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Math Help - Integration Problem

  1. #1
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    Integration Problem

    Okay, so I have the equation (x2-1)cos(3x-x3), and have been asked to integrate said function. We've been given the equation Integral(F1(x)*F(x)n) = ((F(x))n+1)/(n+1)

    If anyone could provide assistance on this, whilst showing the steps involved, it'd be much appreciated.
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  2. #2
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    Re: Integration Problem

    \displaystyle \begin{align*} \int{ \left( x^2 - 1 \right) \cos{ \left( 3x - x^3 \right) } \,dx} &= -\frac{1}{3} \int{ \left( 3 - 3x^2 \right) \cos{ \left( 3x - x^3 \right) } \,dx} \end{align*}

    Now make the substitution \displaystyle \begin{align*} u = 3x - x^3 \implies du = 3 - 3x^2 \, dx \end{align*} and the integral becomes

    \displaystyle \begin{align*} -\frac{1}{3} \int{ \left( 3 - 3x^2 \right) \cos{ \left( 3x - x^3 \right) } \,dx} &= -\frac{1}{3} \int{ \cos{(u)}\,du} \\ &= -\frac{1}{3} \sin{(u)} + C \\ &= -\frac{1}{3} \sin{ \left( 3x - x^3 \right) } + C \end{align*}
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