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

:$\ds \int_0^1 \frac {\arcsin x \arccos x } x \rd x = \dfrac 7 8 \map \zeta 3$
where $\map \zeta 3$ is Apéry's constant: the Riemann $\zeta$ function of $3$.

Proof:
{{begin-eqn}}
{{eqn	| l = \int_0^1 \frac {\arcsin x \arccos x} x \rd x
	    | r = \int_0^1 \frac {\arcsin x \paren {\dfrac \pi 2 - \arcsin x} } x \rd x
	    | c = Sum of Arcsine and Arccosine: $\arcsin x + \arccos x = \dfrac \pi 2$
}}
{{eqn	| r = \frac \pi 2 \int_0^1 \frac {\arcsin x} x \rd x - \int_0^1 \frac {\paren {\arcsin x}^2} x \rd x
	    | c = Linear Combination of Integrals
}}
{{eqn	| r = \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}
	    | c = $x \to \sin x$ and $\rd x \to \cos x \rd x$
}}
{{eqn	| r = \int_0^{\frac \pi 2} x \paren {\frac \pi 2 - x} \cot x \rd x
	    | c = {{Defof|Real Cotangent Function}}
}}
{{end-eqn}}
With a view to expressing the primitive in the form:
{{begin-eqn}}
{{eqn | l = \int u \frac {\d v} {\d x} \rd x
      | r = u v - \int v \frac {\d u} {\d x} \rd x
      | c = Integration by Parts
}}
{{end-eqn}}
let:
{{begin-eqn}}
{{eqn | l = u
      | r = x \paren {\frac \pi 2 - x}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \paren {\frac \pi 2 - 2 x}
      | c = Power Rule for Derivatives
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \cot x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \ln \size {\sin x}
      | c = Primitive of Cotangent Function
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} x \paren {\frac \pi 2 - x} \cot x \rd x
      | r = \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
      | c = Integration by Parts
}}
{{eqn | r = 0 - \int_0^{\frac \pi 2} \paren {\frac \pi 2 - 2 x} \ln \size {\sin x} \rd x
      | c = 
}}
{{eqn | r = -\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
      | c = Fourier Series for $\map \ln {\sin x}$ over $0$ to $\pi$
}}
{{eqn | r = \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
      | c = Linear Combination of Integrals
}}
{{eqn | r = \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
      | c = Primitive of Power
}}
{{eqn | r = 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
      | c = 
}}
{{eqn | r = \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
      | c = Fubini's Theorem
}}
{{eqn | r = \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}
      | c = Linear Combination of Integrals
}}
{{eqn | r = \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} }
      | c = Primitive of $\cos a x$ and Primitive of $x \cos a x$
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac 1 n \paren {-\frac {\map \cos {\pi n} } {2 n^2} + \frac 1 {2 n^2} }
      | c = 
}}
{{eqn | r = \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}
      | c = Cosine of Integer Multiple of Pi
}}
{{eqn | r = \frac 1 2 \map \eta 3 + \frac 1 2 \map \zeta 3
      | c = {{Defof|Dirichlet Eta Function}} and {{Defof|Apéry's Constant}}
}}
{{eqn | r = \frac 1 2 \paren {1 - 2^{1 - 3} } \map \zeta 3 + \frac 1 2 \map \zeta 3
      | c = Riemann Zeta Function in terms of Dirichlet Eta Function: $\map \zeta s = \dfrac 1 {1 - 2^{1 - s} } \map \eta s$
}}
{{eqn | r = \frac 3 8 \map \zeta 3 + \frac 1 2 \map \zeta 3
      | c = 
}}
{{eqn | r = \frac 7 8 \map \zeta 3
      | c = 
}}
{{end-eqn}}
{{qed}}