I'm trying to solve with Mathematica the following integral for $m$ a positive integer

$$s=\int_{0}^{1/2}\frac{\sin^4(\pi\nu m)}{m^2\sin^2(\pi\nu)}\mathrm{d}\nu$$

which should yield $1/(4m)$. If I solve any specific instance, I get the right result, e.g.

Integrate[Sin[Pi \[Nu] 128]^4/(128 Sin[Pi \[Nu]])^2, {\[Nu], 0, 1/2}]

yields $1/512$. However, if I try to solve the general case with

Integrate[Sin[Pi \[Nu] m]^4/(m Sin[Pi \[Nu]])^2, {\[Nu], 0, 1/2}, Assumptions -> m \[Element] PositiveIntegers]

or with

Integrate[Sin[Pi \[Nu] m]^4/(m Sin[Pi \[Nu]])^2, {\[Nu], -1/2, 1/2}, PrincipalValue->True, Assumptions -> m \[Element] PositiveIntegers]

I get a more convoluted expression that yields infinities when substituting an integer value for $m$.

Any tips on how to improve the solution?

I'm trying to solve with Mathematica the following integral for $m$ a positive integer

$$s=\int_{0}^{1/2}\frac{\sin^4(\pi\nu m)}{m^2\sin^2(\pi\nu)}\mathrm{d}\nu$$

which should yield $1/(4m)$ (I already know the solution but want to practice with Mathematica which I use too sparingly). If I solve any specific instance, I get the right result, e.g.

Integrate[Sin[Pi \[Nu] 128]^4/(128 Sin[Pi \[Nu]])^2, {\[Nu], 0, 1/2}]

yields $1/512$. However, if I try to solve the general case with

Integrate[Sin[Pi \[Nu] m]^4/(m Sin[Pi \[Nu]])^2, {\[Nu], 0, 1/2}, Assumptions -> m \[Element] PositiveIntegers]

or with

Integrate[Sin[Pi \[Nu] m]^4/(m Sin[Pi \[Nu]])^2, {\[Nu], -1/2, 1/2}, PrincipalValue->True, Assumptions -> m \[Element] PositiveIntegers]

I get a more convoluted expression that yields infinities when substituting an integer value for $m$.

I thought that the problem might be due to the singularity at $\nu=0$ and also tried to integrate with a lower limit different from zero and then take the limit afterward, but it didn't work. The same starting from the indefinite integral.

Any tips on how to improve the solution?