let give A_p = ∫_0 ^π 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−1) 3) cslculat A_(0 ) , A_1 , A_2 , A


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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

$l e t i n t e g r a t e b y p a r t s$ $1) A_{p} = {[\frac{1}{n} t^{p} s i n (n t)]}_{0}^{π} - \int_{0}^{π} \frac{p}{n} t^{p - 1} s i n (n t) d x$ $= - \frac{p}{n} \int_{0}^{π} t^{p - 1} s i n (n t) d t$ $= - \frac{p}{n} ({[\frac{- 1}{n} t^{p - 1} c o s (n t)]}_{0}^{π} - \int_{0}^{π} - \frac{p - 1}{n} t^{p - 2} c o s (n t) d t)$ $= - \frac{p}{n} (\frac{- π^{p - 1} {(- 1)}^{n}}{n} + \frac{p - 1}{n} \int_{0}^{π} t^{p - 2} c o s (n t) d t)$ $= \frac{p}{n^{2}} π^{p - 1} {(- 1)}^{n} - \frac{p (p - 1)}{n^{2}} A_{p - 2} s o$ $A_{p} = \frac{1}{n^{2}} (p π^{p - 1} {(- 1)}^{n} - p (p - 1) A_{p - 2)})$ $2) A_{2 p} = \frac{1}{n^{2}} (2 p π^{2 p - 1} {(- 1)}^{n} - (2 p) (2 p - 1) A_{2 p - 2})$ $3) A_{2 p + 1}^{} = \frac{1}{n^{2}} ((2 p + 1) π^{2 p} {(- 1)}^{n} - 2 p (2 p + 1) A_{2 p - 1})$ $4) A_{0} = \int_{0}^{π} c o s (n x) d x = {[\frac{1}{n} s i n (n x)]}_{0}^{π} = 0$ $A_{1} = \int_{0}^{π} t c o s (n t) t = {[\frac{t}{n} s i n (n t)]}_{0}^{π} - \int_{0}^{π} \frac{1}{n} s i n (n t) d t$ $= - \frac{1}{n} \int_{0}^{π} s i n (n t) d t = \frac{1}{n^{2}} {[c o s (n t)]}_{0}^{π} = \frac{1}{n^{2}} ({(- 1)}^{n} - 1)$ $A_{2} = \frac{1}{n^{2}} (2 π {(- 1)}^{n} - 2 A_{0}) = \frac{2 π}{n^{2}} {(- 1)}^{n} . . . . b e c o n t u n u e d . . .$
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

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