Editing Fubini's theorem (section)

== Fubini's theorem in multiplications of integrals ==

=== Product of two integrals ===

For the product of two integrals with lower limit zero and a common upper limit we have the following formula:
:{| class = "wikitable"
|<math>\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr] = \int_{0}^{1} \int_{0}^{u} x\,v(xy) \,w(x) + x\,v(x) \,w(xy) \,\mathrm{d}x \,\mathrm{d}y </math>
|}

=== Proof ===

Let <math> V(x) </math> and <math> W(x) </math> are primitive functions of the functions <math> v(x) </math> and <math> w(x ) </math> respectively, which pass through the origin:

:<math>\int_{0}^{u} v(x) \,\mathrm{d}x = V(u), 
\quad \quad 
\int_{0}^{u} w(x) \,\mathrm{d}x = W(u)</math>

Therefore, we have

<math display="block">\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr] = V(u) W(u) </math>

By the [[product rule]], the derivative of the right-hand side is

<math display="block">\frac{\mathrm{d}}{\mathrm{d}x} \bigl[V(x) W(x)\bigr] = V(x)w(x) + v(x)W(x)</math>

and by integrating we have:

<math display="block">\int_{0}^{u} V(x)w(x) + v(x)W(x) \,\mathrm{d}x=V(u) W(u) </math>

Thus, the equation from the beginning we get:

<math display="block">\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{u} V(x)w(x) + v(x)W(x) \,\mathrm{d}x </math>

Now, we introduce a second integration parameter <math>y</math> for the description of the antiderivatives <math>V(x)</math> and <math>W(x)</math>:

: <math>\int_{0}^{1} x\,v(xy) \,\mathrm{d}y = \biggl[V(xy)\biggr]_{y = 0}^{y = 1} = V(x) </math>

: <math>\int_{0}^{1} x\,w(xy) \,\mathrm{d}y = \biggl[W(xy)\biggr]_{y = 0}^{y = 1} = W(x) </math>

By insertion, a double integral appears:

<math display="block">\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{u} \biggl[\int_{0}^{1} x\,v(xy) \,\mathrm{d}y\biggr]w(x) + v(x)\biggl[\int_{0}^{1} x\,w(xy) \,\mathrm{d}y\biggr] \,\mathrm{d}x </math>

Functions that are foreign to the concerned integration parameter can be imported into the inner function as a factor:

<math display="block">\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{u} \biggl[\int_{0}^{1} x\,v(xy) \,w(x) \,\mathrm{d}y\biggr] + \biggl[\int_{0}^{1} x\,v(x)\,w(xy) \,\mathrm{d}y\biggr] \,\mathrm{d}x </math>

In the next step, the [[Integral#Properties|sum rule]] is applied to the integrals:

<math display="block">\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{u} \int_{0}^{1} x\,v(xy) \,w(x) + x\,v(x)\,w(xy) \,\mathrm{d}y \,\mathrm{d}x </math>

And finally, we use the '''Fubini theorem'''

<math display="block">\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{1} \int_{0}^{u} x\,v(xy) \,w(x) + x\,v(x)\,w(xy) \,\mathrm{d}x \,\mathrm{d}y </math>