Integration problem using Integration by Parts

[chwala]

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...

 

[fresh_42]

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

 

[chwala]

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

 

[fresh_42]

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]

$I_n$ 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 $n$.