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Question Number 67795 by mathmax by abdo last updated on 31/Aug/19

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

1) c a l c u l a t e A_{0}, A_{1}, A_{2}

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

Commented by mathmax by abdo last updated on 03/Sep/19

1) 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} x c o s (n x) d x b y p a r t s

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

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

A_{2} = \int_{0}^{π} x^{2} c o s (n x) d x =_{b y p a r t s} {[\frac{x^{2}}{n} s i n (n x)]}_{0}^{π} - \int_{0}^{π} \frac{2 x}{n} s i n (n x) d x

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

= - \frac{1}{n} {- \frac{1}{n} (π {(- 1)}^{n}) + \frac{1}{n^{2}} {[s i n (n x)]}_{0}^{π}}

= \frac{π}{n^{2}} {(- 1)}^{n}

Commented by mathmax by abdo last updated on 04/Sep/19

2) b y p a r t s u^{^{'}} = x^{p} a n d v = c o s (n x) \Rightarrow

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

= \frac{π^{p + 1}}{p + 1} {(- 1)}^{n} + \frac{n}{p + 1} \int_{0}^{π} x^{p + 1} s i n (n x) d x a g a i n b y p a r t s

u^{^{'}} = x^{p + 1} a n d v = s i n (n x) \Rightarrow

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

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

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

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

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

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

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