Definite Integral from 0 to 1 of Arcsine of x by Arccosine of x over x

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Theorem

$\ds \int_0^1 \frac {\arcsin x \arccos x } x \rd x = \dfrac 7 8 \map \zeta 3$

where $\map \zeta 3$ is Apéry's constant: the Riemann $\zeta$ function of $3$.


Proof

\(\ds \int_0^1 \frac {\arcsin x \arccos x} x \rd x\) \(=\) \(\ds \int_0^1 \frac {\arcsin x \paren {\dfrac \pi 2 - \arcsin x} } x \rd x\) Sum of Arcsine and Arccosine: $\arcsin x + \arccos x = \dfrac \pi 2$
\(\ds \) \(=\) \(\ds \frac \pi 2 \int_0^1 \frac {\arcsin x} x \rd x - \int_0^1 \frac {\paren {\arcsin x}^2} x \rd x\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds \frac \pi 2 \int_0^{\frac \pi 2} \frac x {\sin x} \paren {\cos x \rd x} - \int_0^{\frac \pi 2} \frac {x^2} {\sin x} \paren {\cos x \rd x}\) $x \to \sin x$ and $\rd x \to \cos x \rd x$
\(\ds \) \(=\) \(\ds \int_0^{\frac \pi 2} x \paren {\frac \pi 2 - x} \cot x \rd x\) Definition of Real Cotangent Function


With a view to expressing the primitive in the form:

\(\ds \int u \frac {\d v} {\d x} \rd x\) \(=\) \(\ds u v - \int v \frac {\d u} {\d x} \rd x\) Integration by Parts


let:

\(\ds u\) \(=\) \(\ds x \paren {\frac \pi 2 - x}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \paren {\frac \pi 2 - 2 x}\) Power Rule for Derivatives


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \cot x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \ln \size {\sin x}\) Primitive of Cotangent Function


Then:

\(\ds \int_0^{\frac \pi 2} x \paren {\frac \pi 2 - x} \cot x \rd x\) \(=\) \(\ds \intlimits {x \paren {\frac \pi 2 - x} \ln \size {\sin x} } 0 {\frac \pi 2} - \int_0^{\frac \pi 2} \paren {\frac \pi 2 - 2 x} \ln \size {\sin x} \rd x\) Integration by Parts
\(\ds \) \(=\) \(\ds 0 - \int_0^{\frac \pi 2} \paren {\frac \pi 2 - 2 x} \ln \size {\sin x} \rd x\)
\(\ds \) \(=\) \(\ds -\int_0^{\frac \pi 2} \paren {\frac \pi 2 - 2 x} \paren {-\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n} \rd x\) Fourier Series for $\map \ln {\sin x}$ over $0$ to $\pi$
\(\ds \) \(=\) \(\ds \ln 2 \int_0^{\frac \pi 2} \paren {\frac \pi 2 - 2 x} \rd x + \int_0^{\frac \pi 2} \paren {\frac \pi 2 - 2 x} \paren {\sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n} \rd x\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds \ln 2 \intlimits {\frac \pi 2 x - x^2} 0 {\frac \pi 2} + \int_0^{\frac \pi 2} \paren {\frac \pi 2 - 2 x} \paren {\sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n} \rd x\) Primitive of Power
\(\ds \) \(=\) \(\ds 0 + \int_0^{\frac \pi 2} \paren {\frac \pi 2 - 2 x} \paren {\sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n} \rd x\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 n \int_0^{\frac \pi 2} \paren {\frac \pi 2 - 2 x} \cos 2 n x \rd x\) Fubini's Theorem
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 n \paren {\frac \pi 2 \int_0^{\frac \pi 2} \cos 2 n x \rd x - 2 \int_0^{\frac \pi 2} x \cos 2 n x \rd x}\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 n \paren {\frac \pi 2 \intlimits {\frac {\sin 2 n x} {2 n} } 0 {\frac \pi 2} - 2 \intlimits {\frac {\cos 2 n x} {4 n^2} + \frac {x \sin 2 n x} {2 n} } 0 {\frac \pi 2} }\) Primitive of $\cos a x$ and Primitive of $x \cos a x$
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 n \paren {-\frac {\map \cos {\pi n} } {2 n^2} + \frac 1 {2 n^2} }\)
\(\ds \) \(=\) \(\ds \frac 1 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^3} + \frac 1 2 \sum_{n \mathop = 1}^\infty \frac 1 {n^3}\) Cosine of Integer Multiple of Pi
\(\ds \) \(=\) \(\ds \frac 1 2 \map \eta 3 + \frac 1 2 \map \zeta 3\) Definition of Dirichlet Eta Function and Definition of Apéry's Constant
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {1 - 2^{1 - 3} } \map \zeta 3 + \frac 1 2 \map \zeta 3\) Riemann Zeta Function in terms of Dirichlet Eta Function: $\map \zeta s = \dfrac 1 {1 - 2^{1 - s} } \map \eta s$
\(\ds \) \(=\) \(\ds \frac 3 8 \map \zeta 3 + \frac 1 2 \map \zeta 3\)
\(\ds \) \(=\) \(\ds \frac 7 8 \map \zeta 3\)

$\blacksquare$


Sources

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