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  "body": "\n## Summary\n\nThe trigonometric integrals are the sine integral $\\operatorname{Si}(x) = \\int_0^x \\frac{\\sin t}{t}\\, dt$ and the cosine integral $\\operatorname{Ci}(x) = -\\int_x^\\infty \\frac{\\cos t}{t}\\, dt$, with their hyperbolic versions $\\operatorname{Shi}$ and $\\operatorname{Chi}$.\nThey are antiderivatives of $\\frac{\\sin x}{x}$ and $\\frac{\\cos x}{x}$, which have no elementary antiderivatives.\nThe sine integral tends to $\\frac{\\pi}{2}$, the value of the [[Dirichlet integral|0GZNSHMG4Q]], and its overshoot describes the Gibbs phenomenon of Fourier series.\nThey are closely related to the [[exponential integral|EV6ZDQ0NDY]].\n\n## Explanations\n\n### L1 human\n\nThe [[sinc function|P1T98BECWR]] $\\frac{\\sin t}{t}$ has a tall hump at $0$ and smaller and smaller ripples.\nThe sine integral adds up its area from $0$ to $x$.\nIt first rises above $\\frac{\\pi}{2}$, then swings below, and settles at $\\frac{\\pi}{2}$ with smaller and smaller swings.\nIts first peak, at $x = \\pi$, is its largest value, about $1.852$, roughly $18\\%$ above the limit.\n",
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