1. ## Integration Reduction; Formulae

Hi , I am stuck on one line in the solution attached . I am OK to the third line in the mark scheme but
do not understand how ....n a ^(n-1) has been evaluated in the * line .

Thanks

2. ## Re: Integration Reduction; Formulae

Originally Posted by minicooper58
Hi , I am stuck on one line in the solution attached . I am OK to the third line in the mark scheme but
do not understand how ....n a ^(n-1) has been evaluated in the * line .
$\displaystyle \left[-n(a-x)^{n-1}\cos x\right]_0^a$

$\displaystyle =\left[-n(a-a)^{n-1}\cos a\right] - \left[-n(a-0)^{n-1}\cos 0\right]$

$\displaystyle =0 + na^{n-1}\cos 0$

$\displaystyle =na^{n-1}$

3. ## Re: Integration Reduction; Formulae

Originally Posted by minicooper58
Hi , I am stuck on one line in the solution attached . I am OK to the third line in the mark scheme but
do not understand how ....n a ^(n-1) has been evaluated in the * line .

Thanks
\displaystyle \displaystyle \begin{align*} I_n &= \int_0^a{(a - x)^n\cos{x}\,dx} \end{align*}

Using integration by parts with \displaystyle \displaystyle \begin{align*} u = (a - x)^n \implies du = -n(a-x)^{n-1}\,dx \end{align*} and \displaystyle \displaystyle \begin{align*} dv = \cos{x}\,dx \implies v = \sin{x} \end{align*} and the integral becomes

\displaystyle \displaystyle \begin{align*} \int{(a - x)^n\cos{x}\,dx} &= (a - x)^n\sin{x} - \int{-n(a - x)^{n - 1}\sin{x}\,dx} \\ &= (a - x)^n\sin{x} + n\int{(a - x)^{n-1}\sin{x}\,dx} \end{align*}

Using Integration by parts again with \displaystyle \displaystyle \begin{align*} u = (a - x)^{n - 1} \implies du = -(n - 1)(a - x)^{n - 2}\,dx \end{align*} and \displaystyle \displaystyle \begin{align*} dv = \sin{x}\,dx \implies v = -\cos{x} \end{align*} and the integral becomes

\displaystyle \displaystyle \begin{align*} \int{(a - x)^n\cos{x}\,dx} &= (a - x)^n\sin{x} + n\left[-(a-x)^{n-1}\cos{x} - \int{(n - 1)(a - x)^{n-2}\cos{x}\,dx}\right] \\ &= (a - x)^n\sin{x} - n(a - x)^{n - 1}\cos{x} - n(n - 1)\int{(a - x)^{n-2}\cos{x}} \end{align*}

and evaluating between 0 and a will give you the required result