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

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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 $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{j \mathop = 0}^\infty \frac {\left({-1}\right)^j} {a + j d}\) $\quad$ $\quad$


$\blacksquare$


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