Editing Trigonometric integral (section)

== Cosine integral ==
[[Image:Cosine integral.svg|thumb|Plot of {{math|Ci(''x'')}} for {{math|0 < ''x'' ≤ 8''π''}}]]
The different [[cosine]] integral definitions are
<math display="block">\operatorname{Cin}(x) ~\equiv~ \int_0^x \frac{\ 1 - \cos t\ }{ t }\ \operatorname{d} t ~.</math>

{{math|Cin}} is an [[even and odd functions|even]], [[entire function]]. For that reason, some texts define {{math|Cin}} as the primary function, and derive {{math|Ci}} in terms of {{math|Cin&nbsp;.}}

<math display="block">\operatorname{Ci}(x) ~~\equiv~ -\int_x^\infty \frac{\ \cos t\ }{ t }\ \operatorname{d} t ~</math>
<math display="block">~~ \qquad ~=~~ \gamma ~+~ \ln x ~-~ \int_0^x \frac{\ 1 - \cos t\ }{ t }\ \operatorname{d} t ~</math>

<math display="block">~~ \qquad ~=~~ \gamma ~+~ \ln x ~-~ \operatorname{Cin} x ~</math>
for <math>~\Bigl|\ \operatorname{Arg}(x)\ \Bigr| < \pi\ ,</math> where {{math|''γ'' ≈ 0.57721566490&nbsp;...}} is the [[Euler–Mascheroni constant]]. Some texts use {{math|ci}} instead of {{math|Ci}}. The restriction on {{math|Arg(x)}} is to avoid a discontinuity (shown as the orange vs blue area on the left half of the [[#ci_plot_anchor|plot above]]) that arises because of a [[branch cut]] in the standard [[natural logarithm|logarithm function]] ({{math|ln}}).

{{math|Ci(''x'')}} is the antiderivative of {{math|{{sfrac|cos ''x''| ''x'' }} }} (which vanishes as <math>\ x \to \infty\ </math>). The two definitions are related by
<math display="block">\operatorname{Ci}(x) = \gamma + \ln x - \operatorname{Cin}(x) ~.</math>