Definite Integral of Power of u over 1 + Power of u over 0 to 1

Theorem

 $\displaystyle \int_0^1 \dfrac {u^{a - 1} \rd u} {1 + u^d}$ $=$ $\displaystyle \sum_{j \mathop = 0}^\infty \frac {\left({-1}\right)^j} {a + j d}$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \frac 1 a - \frac 1 {a + d} + \frac 1 {a + 2 d} - \frac 1 {a + 3 d} + \cdots$ $\quad$ $\quad$

where $a, d > 0$.

Proof

 $\displaystyle \int_0^1 \frac {u^{a - 1} \rd u} {1 + u^d}$ $=$ $\displaystyle \int_0^1 \frac {u^{a - 1} \rd u} {1 - \left({-u^d}\right)}$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \int_0^1 u^{a - 1} \sum_{j \mathop = 0}^\infty \left({-1}\right)^j u^{j d} \rd u$ $\quad$ Sum of Infinite Geometric Progression $\quad$ $\displaystyle$ $=$ $\displaystyle \int_0^1 \sum_{j \mathop = 0}^\infty \left({-1}\right)^j u^{a - 1 + j d} \rd u$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{j \mathop = 0}^\infty \left({-1}\right)^j \int_0^1 u^{a - 1 + j d} \rd u$ $\quad$ Fubini's Theorem $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{j \mathop = 0}^\infty \left({-1}\right)^j \left[{\frac { u^{a + j d} } {a + j d} }\right]_0^1$ $\quad$ Primitive of Power, Fundamental Theorem of Calculus $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{j \mathop = 0}^\infty \frac {\left({-1}\right)^j} {a + j d}$ $\quad$ $\quad$

$\blacksquare$