Problem 46066. Evaluate the logarithmic integral

Created by ChrisR

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<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 123px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.5px; transform-origin: 407px 61.5px; vertical-align: baseline; "><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.91667px; transform-origin: 12.0667px 7.91667px; unicode-bidi: normal; "><span style="">The</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://en.wikipedia.org/wiki/Logarithmic_integral_function"><span style=""><span style="">logarithmic integral</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="li(x)" style="width: 30.5px; height: 19px;" width="30.5" height="19"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 184px 7.91667px; transform-origin: 184px 7.91667px; unicode-bidi: normal; "><span style=""> plays a role in number theory because the related function </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="Li(x) = li(x) - li(2)" style="width: 126px; height: 19px;" width="126" height="19"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0667px 7.91667px; transform-origin: 40.0667px 7.91667px; unicode-bidi: normal; "><span style=""> provides an estimate for the number of primes less than x. MATLAB's Symbolic Toolbox has the function</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/help/symbolic/sym.logint.html"><span style=""><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; ">logint</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75px 7.91667px; transform-origin: 75px 7.91667px; unicode-bidi: normal; "><span style="">, but it is not available in basic MATLAB or Cody.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 158.967px 7.91667px; transform-origin: 158.967px 7.91667px; unicode-bidi: normal; "><span style="">Write a function to evaluate the logarithmic integral.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1833px 7.91667px; transform-origin: 29.1833px 7.91667px; unicode-bidi: normal; "><span style="">See also </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46081-set-soldner-s-constant"><span style=""><span style="">Cody Problem 46081 "Set Soldner's constant"</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; "><span style="">.</span></span></div></div></div>

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Problem 46066. Evaluate the logarithmic integral

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