Find-0-pi-f-x-cosxdx-given-that-f-x-1-2-k-1-n-coskx-

Question Number 1780 by 112358 last updated on 23/Sep/15

F i n d

\int_{0}^{π} f (x) c o s x d x

g i v e n t h a t

f (x) = 1 + 2 \underset{k = 1}{\sum^{n}} c o s k x .

Commented by Rasheed Soomro last updated on 23/Sep/15

\int_{0}^{π} f (x) c o s x d x

f (x) = 1 + 2 \underset{k = 1}{\sum^{n}} c o s k x .

- - - - - - - - - - - - - - -

\int_{0}^{π} (1 + 2 \underset{k = 1}{\sum^{n}} c o s k x) c o s x d x

\int_{0}^{π} c o s x d x + 2 \underset{k = 1}{\sum^{n}} \int_{0}^{π} (c o s k x) (c o s x) d x

Answered by 123456 last updated on 24/Sep/15

f (x) = 1 + 2 \underset{k = 1}{\sum^{n}} \cos k x

f (x) \cos x = \cos x + 2 \underset{k = 1}{\sum^{n}} \cos k x \cos x

\underset{0}{\int^{π}} f (x) \cos x d x = \underset{0}{\int^{π}} \cos x + 2 \underset{k = 1}{\sum^{n}} \underset{0}{\int^{π}} \cos k x \cos x d x

\underset{0}{\int^{π}} \cos x d x = \sin π - \sin 0 = 0

\underset{0}{\int^{π}} \cos^{2} x d x = \frac{1}{2} \underset{0}{\int^{π}} d x + \frac{1}{2} \underset{0}{\int^{π}} \cos 2 x d x = \frac{1}{2} π

continue

Commented by 112358 last updated on 29/Sep/15

G r a p h i c a l l y I f o u n d t h e v a l u e

o f t h e i n t e g r a l t o b e a p p r o x i m a t e l y π .

C o n s i d e r t h e i d e n t i t y

c o s k x c o s x = \frac{1}{2} (c o s (k + 1) x + c o s (k - 1) x)

T h e n t h e i n t e g r a l u n d e r t h e

s u m m a t i o n b e c o m e s

\int_{0}^{π} (c o s (k + 1) x + c o s (k - 1) x) d x

= \frac{1}{k + 1} s i n (k + 1) x + \frac{1}{k - 1} s i n (k - 1) x ∣_{0}^{π}

= \frac{s i n (k + 1) π}{k + 1} + \frac{s i n (k - 1) π}{k - 1}

W i t h k \in Z^{+} a n d k ⩾ 1, s i n (k + 1) π = 0

a n d s i n (k - 1) π = 0 . B u t \frac{s i n (k - 1) π}{k - 1}

i s u n d e f i n e d a t k = 1 .

∴ \int_{0}^{π} f (x) c o s x d x = π \underset{k = 1}{\sum^{n}} \frac{s i n (k - 1) π}{(k - 1) π}

I f k w e r e a c o n t i n u o u s v a r i a b l e

w e g e t t h a t \lim_{k \to 1} \frac{s i n (k - 1) π}{(k - 1) π} = 1

s o t h a t \int_{0}^{π} f (x) c o s x d x = π

b u t I {d o n}^{'} t f e e l c o m f o r t a b l e w i t h

t h i s a p p r o a c h s i n c e k i s a d i s c r e t e

q u a n t i t y .

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