∫ 积分计算器Integral Calculator

使用说明与常见问题How-to & FAQ

「积分计算器」是什么？What is the Integral Calculator tool?

计算不定积分（原函数）和定积分：支持多项式、三角、指数、对数、有理函数等，定积分给出数值结果，找不到初等原函数时会明确提示。Compute indefinite integrals (antiderivatives) and definite integrals for polynomials, trig, exponential, logarithmic and rational functions; definite integrals give a numeric value, and it tells you clearly when no elementary antiderivative is found.

如何使用「积分计算器」？How do I use the Integral Calculator tool?

在「积分计算器」中，按需设置「被积函数 f(x)」、「变量」、「类型」、「下限 a」、「上限 b」等选项，然后点击「计算」按钮即可。In the Integral Calculator: Set "Integrand f(x)", "Variable", "Type", "Lower a", "Upper b", then click "Integrate".

不定积分为什么要加 C？Why add + C?

原函数加上任意常数后导数不变，所以不定积分的结果是一族函数，用 + C 表示积分常数。Adding any constant doesn't change the derivative, so an indefinite integral is a family of functions; + C is the constant of integration.

定积分的结果精确吗？How accurate are definite integrals?

有原函数时按牛顿–莱布尼茨公式计算，否则用数值积分；结果保留约 12 位有效数字。会扫描区间内的奇点：像 ∫₀¹ 1/x 这样不收敛的反常积分会提示“发散”，可积奇点（如 1/√x）会给出精度提醒；小于 1e-10 的舍入误差按 0 显示。With an antiderivative it uses the fundamental theorem of calculus, otherwise numerical integration; results keep about 12 significant digits. The interval is scanned for singularities: a divergent improper integral such as ∫₀¹ 1/x is reported as divergent, an integrable singularity (like 1/√x) gets an accuracy caution, and round-off below 1e-10 is shown as 0.

支持哪些写法？What syntax is supported?

四则运算 + - * /、乘方 ^、括号，函数 sin cos tan asin acos atan sinh cosh tanh sqrt abs exp log（自然对数，ln 也可以）等，常数 pi 和 e。乘号可以省略（2x、3sin(x)），也可以用 ×、÷、π、√。+ - * /, powers with ^, parentheses, functions such as sin cos tan asin acos atan sinh cosh tanh sqrt abs exp log (natural log; ln also works), and the constants pi and e. Multiplication can be implicit (2x, 3sin(x)); ×, ÷, π and √ are accepted too.

用的是什么计算引擎？What engine does it use?

开源符号计算库 nerdamer（Apache-2.0 许可），在独立的后台线程（Web Worker）中运行，超过 10 秒会自动停止，不会卡住页面；所有计算都在本地完成。The open-source symbolic library nerdamer (Apache-2.0), running in a background Web Worker that stops after 10 seconds so the page never freezes; everything is computed locally.

「积分计算器」免费吗？需要注册吗？Is the Integral Calculator free? Do I need an account?

「积分计算器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Integral Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.