Integration Problem

**Skorm178** · July 8th 2013, 11:43 PM

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.

**Prove It** · July 9th 2013, 12:35 AM

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

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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