Editing Integration by parts (section)

===Product of two functions===
The theorem can be derived as follows. For two [[continuously differentiable]] [[function (mathematics)|functions]] <math>u(x)</math> and <math>v(x)</math>, the [[product rule]] states:

<math display="block">\Big(u(x)v(x)\Big)'  = u'(x) v(x) + u(x) v'(x).</math>

Integrating both sides with respect to <math>x</math>,

<math display="block">\int \Big(u(x)v(x)\Big)'\,dx  = \int u'(x)v(x)\,dx + \int u(x)v'(x) \,dx, </math>

and noting that an [[indefinite integral]] is an antiderivative gives

<math display="block">u(x)v(x)  = \int u'(x)v(x)\,dx + \int u(x)v'(x)\,dx,</math>

where we neglect writing the [[constant of integration]]. This yields the formula for '''integration by parts''':

<math display="block">\int u(x)v'(x)\,dx  = u(x)v(x) - \int u'(x)v(x) \,dx, </math>

or in terms of the [[differentials of a function|differentials]] <math> du=u'(x)\,dx</math>, <math>dv=v'(x)\,dx, \quad</math>

<math display="block">\int u(x)\,dv  = u(x)v(x) - \int v(x)\,du.</math>

This is to be understood as an equality of functions with an unspecified constant added to each side. Taking the difference of each side between two values <math>x = a</math> and <math>x = b</math> and applying the [[fundamental theorem of calculus]] gives the definite integral version:
<math display="block"> \int_a^b u(x) v'(x) \, dx 
    = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx . </math>
The original integral <math>\int uv' \, dx</math> contains the [[derivative]] {{mvar|v'}}; to apply the theorem, one must find {{mvar|v}}, the [[antiderivative]] of {{mvar|v'}}, then evaluate the resulting integral <math>\int vu' \, dx.</math>