Find this integration with conditions

Question

$$\int_{0}^{\tau} t\,(t-t^2)^{n-0.5}dt$$ (where$\ n\gt 0.5 $, $0\le t \le \tau$)

i am evaluating an inner product for the shifted Ultraspherical polynomials at zero degrees and what i reached to is this integration of the weight function with respect to $t$ of these polynomials multiplied by $t$ i am not sure if i should provide the whole picture, but i think it's a good idea to tell that this integration is derived from $$(\int_{0}^{t}\psi_0{(t)}dt,\,\psi_0(t))_{w(t)}$$ the inner product with respect to the weight function of the Shifted Ultraspherical polynomials. Where $$\psi_0(t)=1$$ i am not sure with my tags but i hope it's useful.

Answer 1

Factoring a bit gives

$$\int_0^\tau t^{n+1/2}(1-t)^{n-1/2}~\mathrm dt=B\left(\tau;n+\frac32,n+\frac12\right)$$

where $B$ is the incomplete beta function. There is a notable issue that when $\tau\notin[0,1]$ we get a complex integral.

For natural $n$ and $\tau\in[0,1]$ it is possible to solve with the trigonometric substitution of $t=\sin^2(\theta)$ to get

$$\int_0^{\arcsin(\sqrt\tau)}2\sin^{2n+2}(\theta)\cos^{2n}(\theta)~\mathrm d\theta$$

which can be tackled using standard techniques.