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

## Theorem

 $\ds \int_0^1 \dfrac {u^{a - 1} \rd u} {1 + u^d}$ $=$ $\ds \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {a + j d}$ $\ds$ $=$ $\ds \frac 1 a - \frac 1 {a + d} + \frac 1 {a + 2 d} - \frac 1 {a + 3 d} + \cdots$

where $a, d > 0$.

## Proof

 $\ds \int_0^1 \frac {u^{a - 1} \rd u} {1 + u^d}$ $=$ $\ds \int_0^1 \frac {u^{a - 1} \rd u} {1 - \paren {-u^d} }$ $\ds$ $=$ $\ds \int_0^1 u^{a - 1} \sum_{j \mathop = 0}^\infty \paren {-1}^j u^{j d} \rd u$ Sum of Infinite Geometric Sequence $\ds$ $=$ $\ds \int_0^1 \sum_{j \mathop = 0}^\infty \paren {-1}^j u^{a - 1 + j d} \rd u$ $\ds$ $=$ $\ds \sum_{j \mathop = 0}^\infty \paren {-1}^j \int_0^1 u^{a - 1 + j d} \rd u$ Fubini's Theorem $\ds$ $=$ $\ds \sum_{j \mathop = 0}^\infty \paren {-1}^j \intlimits {\frac {u^{a + j d} } {a + j d} } 0 1$ Primitive of Power, Fundamental Theorem of Calculus $\ds$ $=$ $\ds \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {a + j d}$

$\blacksquare$