# Integration problem using Integration by Parts

Gold Member

## 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...

i would appreciate alternative method...

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fresh_42
Mentor
You can differentiate to check it, but watch out for the integration limits, as differentiation isn't exactly the opposite of integration.

Gold Member
hi fresh...long time...ok you may direct me...i am trying to refresh in this things...

fresh_42
Mentor
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.

pasmith
Homework Helper
$I_n$ is the imaginary part of $$J_n = \int_{1/2}^1 x^{-n} \mathrm{e}^{\mathrm{i} \pi x}\,dx$$ and you can establish the result* by interating this by parts once to get a relation between $J_{n+2}$ and $J_{n+1}$, and you can then substitute for $J_{n+1}$ in terms of $J_n$ before taking imaginary parts.

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

SammyS