I am trying to compute the integral $\int_0^1 p^x(1-p)^{n-x+1} dp$ using integration by parts, however, I am stuck in a loop where I keep computing integrals of products of polynomials (of higher order than the originals).
Is there a general method to compute such integrals?
Assuming the exponents are integers the integration by parts should terminate. The standard convention for the exponents in the integral ("Eulerian Integral of the First Kind") would be $\int _0^1 {x^{m-1}(1-x)^{n-1}dx}$. Letting $u=x^{m-1}$ and $dv=(1-x)^{n-1}dx$ for an integration by parts gives $$\int _0^1 {x^{m-1}(1-x)^{n-1}dx}= \frac{m-1}{n} \int _0 ^1 {x^{m-2}(1-x)^{n}dx}$$ Proceeding inductively after (m-1) intgegration by parts $$\int _0^1 {x^{m-1}(1-x)^{n-1}dx}= \frac{(m-1)(m-2)\cdot\cdot\cdot2\cdot1}{n(n+1)(n+2)\cdot\cdot\cdot(n+m-2)} \int _0 ^1 {(1-x)^{n+m-2}dx}$$ After evaluating the last integral $$\int _0^1 {x^{m-1}(1-x)^{n-1}dx}= \frac{(m-1)(m-2)\cdot\cdot\cdot2\cdot1}{n(n+1)(n+2)\cdot\cdot\cdot(n+m-1)}=\frac{(m-1)!\space (n-1)!}{(n+m-1)!}$$ The latter expression defines the Beta function $B (m,n)$.