Question

integration from 0 to 1
of (m^(m-1)((1-x)^n-1)/ ((a+x)^(m+n)

Answer 1

Conditions

integration from 0 to 1

of $(m^{\wedge}(m - 1)((1 - x)^{\wedge}n - 1) / ((a + x)^{\wedge}(m + n))$

Solution

$$\begin{array}{l} \int \frac {m ^ {m - 1} ((1 - x) ^ {n} - 1)}{(a + x) ^ {m + n}} d x = \\ \frac {1}{m + n - 1} m ^ {m - 1} \left(\frac {1 - x}{a + 1}\right) ^ {- n} (a + x) ^ {- m - n + 1} \left(\left(\frac {1 - x}{a + 1}\right) ^ {n} - \right. \\ (1 - x) ^ {n} _ {2} F _ {1} \left(- m - n + 1, - n; - m - n + 2; \frac {a + x}{a + 1}\right) + \text{constant} \\ \end{array}$$ $$\begin{array}{l} \frac {m ^ {m - 1} (a + x) ^ {- m - n + 1}}{m + n - 1} - \frac {1}{m + n - 1} m ^ {m - 1} (a + 1) ^ {n} \\ (a + x) ^ {- m - n + 1} _ {2} F _ {1} \left(- m - n + 1, - n; - m - n + 2; \frac {a + x}{a + 1}\right) \\ \exp \left(- 2 i \pi n \left[ \frac {\arg (a + 1)}{2 \pi} - \frac {\arg (1 - x)}{2 \pi} + \frac {1}{2} \right]\right) + \text{constant} \\ \end{array}$$