Integration problem using Integration by Parts

  • Thread starter chwala
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  • #1
chwala
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Homework Statement:

The integral ##I_{n}##, where ##n## is positive is given by ##I_{n}=\int_{0.5}^1x^{-n} sinπx\, dx##

show that,

##n(n+1)I_{n+2}=2^{n+1} + π-π^2 I_{n}##

Relevant Equations:

integration by parts...
1594298971707.png


i would appreciate alternative method...
 

Answers and Replies

  • #2
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You can differentiate to check it, but watch out for the integration limits, as differentiation isn't exactly the opposite of integration.
 
  • #3
chwala
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hi fresh...long time...ok you may direct me...i am trying to refresh in this things...
 
  • #4
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Let's define ##f(x)=n(n+1)I_{n+2}-2^{n+1}+\pi - \pi^2 I_n##. Then show that ##f'(x)=0## which means ##x \longmapsto f(x)## is constant, say ##f(x)=C##. At last calculate ##C## from the definition of ##f(x)##, or show that only ##C=0## is possible.

I haven't done it. so I don't know how it will work. The entire question is a standard example for integration by parts. Another possibility could be using a Weierstraß substitution, but I don't think this would change a lot.
 
  • #5
pasmith
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[itex]I_n[/itex] is the imaginary part of [tex]
J_n = \int_{1/2}^1 x^{-n} \mathrm{e}^{\mathrm{i} \pi x}\,dx
[/tex] and you can establish the result* by interating this by parts once to get a relation between [itex]J_{n+2}[/itex] and [itex]J_{n+1}[/itex], and you can then substitute for [itex]J_{n+1}[/itex] in terms of [itex]J_n[/itex] before taking imaginary parts.

*So far as I can tell, there should be an [itex]n[/itex] multiplying [itex]2^{n+1}[/itex]. chwala's work agrees with this until the last line, where it mysteriously disappears upon multiplying [itex](n+1)I_{n+2}[/itex] by [itex]n[/itex].
 
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