Definite Integral from 0 to 1 of Power of u over 1 + Power of u
Jump to navigation
Jump to search
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {a + j d}\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.34$