let-give-A-p-0-pi-t-p-cos-nx-with-nand-p-from-N-1-find-a-relation-between-A-p-and-A-p-2-2-find-arelation-between-A-2p-and-A-2p-2-3-find-a-relation-betweer-A-2p-1-and-A-2p

Question Number 28071 by abdo imad last updated on 20/Jan/18

l e t g i v e A_{p} = \int_{0}^{π} t^{p} c o s (n x) w i t h n a n d p f r o m N

1) f i n d a r e l a t i o n b e t w e e n A_{p} a n d A_{p - 2}

2) f i n d a r e l a t i o n b e t w e e n A_{2 p} a n d A_{2 p - 2}

3) f i n d a r e l a t i o n ? b e t w e e r A_{2 p + 1} a n d A_{2 p - 1}

3) c s l c u l a t A_{0}, A_{1}, A_{2}, A_{2} .

Commented by abdo imad last updated on 26/Jan/18

l e t i n t e g r a t e b y p a r t s

1) A_{p} = {[\frac{1}{n} t^{p} s i n (n t)]}_{0}^{π} - \int_{0}^{π} \frac{p}{n} t^{p - 1} s i n (n t) d x

= - \frac{p}{n} \int_{0}^{π} t^{p - 1} s i n (n t) d t

= - \frac{p}{n} ({[\frac{- 1}{n} t^{p - 1} c o s (n t)]}_{0}^{π} - \int_{0}^{π} - \frac{p - 1}{n} t^{p - 2} c o s (n t) d t)

= - \frac{p}{n} (\frac{- π^{p - 1} {(- 1)}^{n}}{n} + \frac{p - 1}{n} \int_{0}^{π} t^{p - 2} c o s (n t) d t)

= \frac{p}{n^{2}} π^{p - 1} {(- 1)}^{n} - \frac{p (p - 1)}{n^{2}} A_{p - 2} s o

A_{p} = \frac{1}{n^{2}} (p π^{p - 1} {(- 1)}^{n} - p (p - 1) A_{p - 2)})

2) A_{2 p} = \frac{1}{n^{2}} (2 p π^{2 p - 1} {(- 1)}^{n} - (2 p) (2 p - 1) A_{2 p - 2})

3) A_{2 p + 1}^{} = \frac{1}{n^{2}} ((2 p + 1) π^{2 p} {(- 1)}^{n} - 2 p (2 p + 1) A_{2 p - 1})

4) A_{0} = \int_{0}^{π} c o s (n x) d x = {[\frac{1}{n} s i n (n x)]}_{0}^{π} = 0

A_{1} = \int_{0}^{π} t c o s (n t) t = {[\frac{t}{n} s i n (n t)]}_{0}^{π} - \int_{0}^{π} \frac{1}{n} s i n (n t) d t

= - \frac{1}{n} \int_{0}^{π} s i n (n t) d t = \frac{1}{n^{2}} {[c o s (n t)]}_{0}^{π} = \frac{1}{n^{2}} ({(- 1)}^{n} - 1)

A_{2} = \frac{1}{n^{2}} (2 π {(- 1)}^{n} - 2 A_{0}) = \frac{2 π}{n^{2}} {(- 1)}^{n} \dots . b e c o n t u n u e d \dots

Commented by abdo imad last updated on 26/Jan/18

A_{3} = \frac{1}{n^{2}} (3 π^{2} {(- 1)}^{n} - 6 A_{1})

= \frac{1}{n^{2}} (3 π^{2} {(- 1)}^{n} - \frac{6}{n^{2}} ({(- 1)}^{n} - 1))

= \frac{3 π^{2}}{n^{2}} {(- 1)}^{n} - \frac{6}{n^{4}} ({(- 1)}^{n} - 1) .

Commented by abdo imad last updated on 26/Jan/18

A_{p} = \int_{0}^{π} t^{p} c o s (n t) d t .