Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x
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Theorem
- $\ds \int_0^1 x^m \paren {\ln x}^n \rd x = \frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {m + 1}^{n + 1} }$
where:
- $n$ is a non-negative integer
- $m$ is a real number with $m > -1$.
Proof 1
Let:
- $x = \map \exp {-\dfrac u {m + 1} }$
Then, by Derivative of Exponential Function:
- $\dfrac {\d x} {\d u} = -\dfrac 1 {m + 1} \map \exp {-\dfrac u {m + 1} }$
We have by Exponential of Zero:
- as $x \to 1$, $u \to 0$
We also have, by Exponential Tends to Zero and Infinity:
- as $x \to 0$, $u \to \infty$
So:
\(\ds \int_0^1 x^m \paren {\ln x}^n \rd x\) | \(=\) | \(\ds \int_\infty^0 \paren {\map \exp {-\frac u {m + 1} } }^m \paren {-\frac u {m + 1} }^n \paren {-\frac 1 {m + 1} \map \exp {-\frac u {m + 1} } \rd u}\) | Integration by Substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {m + 1} \int_0^\infty \paren {\map \exp {-\frac u {m + 1} } }^m \paren {-\frac u {m + 1} }^n \paren {\map \exp {-\frac u {m + 1} } \rd u}\) | Reversal of Limits of Definite Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {m + 1} \int_0^\infty \map \exp {-u \paren {\frac m {m + 1} + \frac 1 {m + 1} } } \frac {u^n} {\paren {m + 1}^n} \rd u\) | Exponential of Sum, Power of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {\paren {m + 1}^{n + 1} } \int_0^\infty e^{-u} u^{\paren {n + 1} - 1} \rd u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {m + 1}^{n + 1} }\) | Definition of Gamma Function |
$\blacksquare$
Proof 2
From Primitive of Power, we have:
- $\ds \int_0^1 x^m \rd x = \frac 1 {m + 1}$
We have:
\(\ds \frac {\d^n} {\d m^n} \int_0^1 x^m \rd x\) | \(=\) | \(\ds \int_0^1 \frac {\partial^n} {\partial m^n} x^m \rd x\) | Definite Integral of Partial Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 x^m \paren {\ln x}^n \rd x\) | Derivative of Power of Constant |
So:
\(\ds \int_0^1 x^m \paren {\ln x}^n \rd x\) | \(=\) | \(\ds \frac {\d^n} {\d m^n} \paren {\frac 1 {m + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\d^{n + 1} } {\d m^{n + 1} } \paren {\map \ln {m + 1} }\) | Derivative of Natural Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n \map \Gamma {n + 1} } { \paren {m + 1}^{n + 1} }\) | $n$th Derivative of Natural Logarithm |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Logarithmic Functions: $15.90$