Typical Applications

Once you know the definition and derivative property, practice is key. These examples show how “differentiate an upper-limit integral” works in varied settings.

基本应用

例 1：$F(x) = \int_0^x \sin t \, dt$，求 $F'(x)$

解：By $F'(x)=f(x)$, we get $F'(x) = \sin x$.

例 2：$F(x) = \int_1^x \dfrac{1}{t} \, dt$，求 $F'(x)$

解：Treat $f(t)=\frac{1}{t}$, so $F'(x) = \dfrac{1}{x}$.

复杂应用

例 3：$F(x) = \int_0^x e^{-t^2} \, dt$，求 $F'(x)$

解：$F'(x) = e^{-x^2}$. Even without an elementary antiderivative, the derivative is immediate.

例 4：$F(x) = \int_0^x \dfrac{\sin t}{t} \, dt$，求 $F'(x)$

解：$F'(x) = \dfrac{\sin x}{x}$. As long as the integrand is meaningful at the upper limit, the derivative exists.

These examples highlight: the derivative of the upper-limit integral equals the integrand evaluated at the upper limit, regardless of how hard the integral itself is.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$F'(x)$	数学符号	导数	积分上限函数的导数，等于被积函数

中英对照

中文术语	英文术语	音标	说明
积分上限函数	upper-limit integral function	/ˈʌpər ˈlɪmɪt ɪnˈtɛɡrəl ˈfʌŋkʃən/	形如 $F(x)=\int_a^x f(t)\,dt$ 的函数
被积函数	integrand	/ˈɪntəˌɡrænd/	被积分号包围的函数 $f(t)$

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