∫_0 ^π cos^n (x)∙cos (nx)dx=(π/2^n )


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Question Number 140684 by qaz last updated on 11/May/21

$\int_{0}^{π} \cos^{n} (x) \cdot \cos (n x) d x = \frac{π}{2^{n}}$
	Answered by mnjuly1970 last updated on 11/May/21

	$ϕ := R e (\frac{1}{2} \int_{0}^{2 π} {c o s}^{n} (x) e^{i n (x)} d x)$ $:= R e (\frac{1}{2} \int_{0}^{2 π} {(\frac{e^{i x} + e^{- i x}}{2})}^{n} e^{i n x} d x)$ $: \overset{z := e^{i x}}{=} R e \int_{C :∣ z ∣= 1} \frac{1}{2^{n}} {(\frac{z + z^{- 1}}{1})}^{n} z^{n} \frac{d z}{i z}$ $:= R e \frac{1}{2} \int_{C :∣ z ∣= 1} \frac{1}{2^{n}} {(\frac{z^{2} + 1}{1})}^{n} \frac{d z}{i z} = 2 π \frac{1}{2^{n + 1}} (1) = \frac{π}{2^{n}}$
	Commented by mnjuly1970 last updated on 11/May/21

	$t h a n k s a l o t s i r q a z$ $g r a t e f u l . . .$
	Commented by qaz last updated on 11/May/21

	$n i c e s o l u t i o n S i r . I a l s o {d o n}^{'} t l i k e i n d u c t i o n . . .$
	Answered by mathmax by abdo last updated on 11/May/21

	$A_{n} = \int_{0}^{π} \cos^{n} x . \cos (nx) dx \Rightarrow A_{n} = Re (\int_{0}^{π} \cos^{n} x e^{inx} dx) but$ $\int_{0}^{π} \cos^{n} x e^{inx} dx = \int_{0}^{π} {(\frac{e^{ix} + e^{- ix}}{2})}^{n} e^{inx} dx$ $= \frac{1}{2^{n}} \int_{0}^{π} e^{inx} \sum_{k = 0}^{n} C_{n}^{k} {(e^{ix})}^{k} {(e^{- ix})}^{n - k} dx$ $= \frac{1}{2^{n}} \sum_{k = 0}^{n} \int_{0}^{π} C_{n}^{k} e^{inx} e^{ikx} . e^{- inx} e^{ikx} dx$ $= \frac{1}{2^{n}} \sum_{k = 0}^{n} C_{n}^{k} \int_{0}^{π} e^{2 ikx} dx = \frac{π}{2^{n}} + \frac{1}{2^{n}} \sum_{k = 1}^{n} C_{n}^{k} {[\frac{1}{2 ik} e^{2 ikx}]}_{0}^{π}$ $= \frac{π}{2^{n}} + 0 = \frac{π}{2^{n}}$
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